The Book

William Stein August 27, 1997

Contents

1 Algebraic Geometry 2

1.1 Algebraic Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.1 Hilbert’s Nullstellensatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.2 Nonsingularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Glueing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.3 Properties of Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3.1 Connected, irreducible, reduced, integral, affine . . . . . . . . . . . . . . . 3 1.3.2 Geometrically reduced and irreducible . . . . . . . . . . . . . . . . . . . . 4 1.3.3 Locally noetherian and noetherian . . . . . . . . . . . . . . . . . . . . . . 4 1.3.4 Normal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3.5 Regular . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4.1 Operations on sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5 Direct and inverse image functors . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.5.1 Quasi-coherent and coherent sheaves . . . . . . . . . . . . . . . . . . . . . 6 1.5.2 Locally free sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.6 Properties of Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.6.1 Quasi-compact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.6.2 Finite type and locally of finite type . . . . . . . . . . . . . . . . . . . . . 8 1.6.3 Finite and Quasi-finite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.6.4 Affine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.6.5 Immersions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.6.6 Separated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.6.7 Proper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.6.8 Projective and quasi-projective . . . . . . . . . . . . . . . . . . . . . . . . 11 1.6.9 Flat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.6.10 Smooth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.6.11 Étale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.6.12 Unramified . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.7 Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.7.1 Weil divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.7.2 Cartier divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.7.3 Invertible sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.7.4 Picard groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.8 Morphisms to projective space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.9 Basic sheaf cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.10 Algebraic curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.10.1 Dictionary between fields and curves . . . . . . . . . . . . . . . . . . . . . 14 2 Lie Algebras 15

2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.1 Linear lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2.2 Classical lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2.3 Other linear lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2.4 Derivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2.5 Basic Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2.6 Solvability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.2.7 Nilpotence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3 Lie’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3.1 Some remarks on Jordan and rational canonical form. . . . . . . . . . . . 20 2.4 Cartan’s Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.5 The Killing form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.6 Basic facts about representations . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.6.1 Duality and tensor products . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.6.2 Casimir element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.6.3 Weyl’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.7 Representations of 𝔰𝔩(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.8 The root system arising from a toral decomposition . . . . . . . . . . . . . . . . . 23 2.9 Root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3 Algebraic Number Theory 26 3.1 Trace map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.2 The Discriminant and Ramification . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3 The Minkowski Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.4 Cyclotomic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.5 Hilbert’s 90 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.6 Kummer Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.7 Hensel’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.8 Ideles and adeles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.8.1 Restricted products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4 Galois Theory 31 4.1 Separability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

5 Number theory qual review 32

5.1 Discriminant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5.1.2 The module index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5.1.3 Trace form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

6 Homological Algebra 35 6.1 Inverse and Direct Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

7 Galois and Group Cohomology 36 7.1 Order and Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 7.2 H^{1}(GL(V), V) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 7.3 Galois cohomology of abelian varieties . . . . . . . . . . . . . . . . . . . . . . . . 38 7.4 Descent theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 8 Class Field Theory 40

8.1 Main theorems of class field theory . . . . . . . . . . . . . . . . . . . . . . . . . . 40 8.2 Ray and ring class fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 8.3 Tables of class groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

8.3.1 Quadratic imaginary extensions . . . . . . . . . . . . . . . . . . . . . . . . 40 8.3.2 Gauss’ class number problem. . . . . . . . . . . . . . . . . . . . . . . . . . 41 8.3.3 Real quadratic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 8.3.4 Cyclotomic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

9 Math 254B: Final Problem Set 43 9.1 Indexes of Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 9.2 Euler Systems and Indexes of Cohomology Classes . . . . . . . . . . . . . . . . . 48

9.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 9.2.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 9.2.3 A Result About the Euler System . . . . . . . . . . . . . . . . . . . . . . 50 9.2.4 Proofs of Theorems 1 and 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 51

9.3 Order and index questions for H^{1}(ℚ, E) . . . . . . . . . . . . . . . . . . . . . . . 52 9.3.1 Cohomological criterion for order=index . . . . . . . . . . . . . . . . . . . 52 9.3.2 Order=index for elements of � (E/ℚ) . . . . . . . . . . . . . . . . . . . . 52 9.3.3 Cassels’ example that order need not equal index . . . . . . . . . . . . . . 52 9.3.4 Order=index over a local field . . . . . . . . . . . . . . . . . . . . . . . . . 52

9.4 The basic exact sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

10 Elliptic Curves 54

10.1 Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 10.1.1 Cremona’s table for N ≤ 64 . . . . . . . . . . . . . . . . . . . . . . . . . . 55 10.1.2 Big � ’s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 10.1.3 Complex multiplication elliptic curves . . . . . . . . . . . . . . . . . . . . 57

10.2 Curves of high rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 10.3 Weierstrass Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

10.3.1 Finding weierstrass form of an elliptic curve . . . . . . . . . . . . . . . . . 58 10.3.2 Invariants attached to weierstrass models . . . . . . . . . . . . . . . . . . 59

10.4 The Birch and Swinnerton-Dyer Conjecture . . . . . . . . . . . . . . . . . . . . . 60 10.4.1 The L-series attached to an elliptic curve . . . . . . . . . . . . . . . . . . 60 10.4.2 The Birch and Swinnerton-Dyer Conjecture . . . . . . . . . . . . . . . . . 60 10.4.3 The Tamagawa numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 10.4.4 The Néron-Tate canonical height . . . . . . . . . . . . . . . . . . . . . . . 61 10.4.5 The real period of an elliptic curve . . . . . . . . . . . . . . . . . . . . . . 62

10.5 Elliptic Curves over Finite Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 10.5.1 Hasse Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 10.5.2 What values in the Hasse interval are attained? . . . . . . . . . . . . . . . 63

11 Computation 65

11.1 Pari . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 11.1.1 Compiling Pari under Linux . . . . . . . . . . . . . . . . . . . . . . . . . . 65 11.1.2 Library Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 11.1.3 gp Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 11.1.4 Numerical Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

11.2 LiDIA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

11.2.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 11.3 Cremona’s programs: mwrank, etc. . . . . . . . . . . . . . . . . . . . . . . . . . . 67

11.3.1 Bug Report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 11.4 GAP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

11.4.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 12 Unix Notes 70

12.1 Scripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 12.1.1 pack and unpack. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 12.1.2 Check spelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 12.1.3 /etc/profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 12.1.4 Configuring X Windows . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 12.1.5 Trash can . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 12.1.6 Useful Aliases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

12.2 Networking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

12.2.1 Windowing programs over the net. . . . . . . . . . . . . . . . . . . . . . . 71 12.3 TEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

12.3.1 Characters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 12.4 Conditionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

12.4.1 Conditionals in L^{A}TEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 12.4.2 Conditionals in TEX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

12.5 Emacs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

13 Euler Systems 74

13.1 Rubin, The work of Kolyvagin on the arithmetic of elliptic curves . . . . . . . . . 74 13.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 13.1.2 Group Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 13.1.3 Application of Tate Local Duality . . . . . . . . . . . . . . . . . . . . . . 75 13.1.4 The Euler System of Heegner points . . . . . . . . . . . . . . . . . . . . . 75 13.1.5 Cohomology Classes Arising from Heegner points . . . . . . . . . . . . . . 75 13.1.6 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

13.2 Gross, Kolyvagin’s work on modular elliptic curves . . . . . . . . . . . . . . . . . 79 13.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 13.2.2 Constructing the Euler System . . . . . . . . . . . . . . . . . . . . . . . . 80 13.2.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

13.3 Mccallum, Kolyvagin’s work on Shafarevich-Tate groups . . . . . . . . . . . . . . 81

14 Commutative Algebra 82 14.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 14.2 Basic Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 14.3 Module theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

14.3.1 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 14.3.2 Projective, injective, and flat modules . . . . . . . . . . . . . . . . . . . . 83

14.4 Étale extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 14.5 Local Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

14.5.1 Regular Local Rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 14.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 14.7 Associated primes and primary decomposition . . . . . . . . . . . . . . . . . . . . 85 14.8 Going up and going down . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 14.9 Discriminants and resultants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

15 Iwasawa Theory 88

15.1 The Main Conjecture Of Iwasawa Theory . . . . . . . . . . . . . . . . . . . . . . 88 15.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 15.1.2 Recall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 15.1.3 The connection between F^{×}/(F^{×})^{M} and the Iwasawa algebra . . . . . . . 89 15.1.4 The Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 15.1.5 Choosing cunning primes . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 15.1.6 More Is True . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 15.1.7 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 16 Group Theory 92

16.1 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

16.1.1 Characteristic Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 16.2 Group Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 16.3 The Sylow Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

16.3.1 p-groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 16.4 Composition Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 16.5 Solvable Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

16.5.1 Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 16.6 Nilpotent Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

17 Group Cohomology 97 17.1 The Definition of Group Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . 97 17.2 Morphisms of Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

17.2.1 The category P of pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 17.2.2 Inf, Res, Inner Automorphisms, Shapiro’s Lemma . . . . . . . . . . . . . 98 17.2.3 Bimodules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 17.2.4 Shapiro’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

17.3 Sylow p-subgroups and Sharpness . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 17.4 The case G ⊂ Aut M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

17.4.1 Application to galois representations . . . . . . . . . . . . . . . . . . . . . 101 17.5 Group extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

18 English 102 18.1 Spelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 18.2 Grammar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

19 Elementary Number Theory 103 19.0.1 Density of square free integers . . . . . . . . . . . . . . . . . . . . . . . . 103

20 Analysis and Measure Theory 104

Preface

This is my book in which I store the important mathematical results I use. Optimally, nothing should be written in here unless it is completely correct. To help insure this I have laid out the following guidelines.

1. When a new entry is added it must be put in an unverified environment.

2. After having been carefully re-read (at least one day later) the environment can be upgraded to a verified environment.

3. After the statement has been independently checked the environment can be changed to

correct.

4. Each chapter is contained in a separate file whose name is ch subjectname.tex.

William Stein was@math.berkeley.edu Berkeley, California

Chapter 1

Algebraic Geometry

********* Begin Unverified *********

1.1 Algebraic Varieties

Throughout this section k is an arbitrary algebraically closed field.

1.1.1 Hilbert’s Nullstellensatz

Lemma 1.1.1. If K is a field which is finitely generated as an algebra over k, then K is algebraic over k.

Proof. The proof relies on the existence of infinitely primes in a polynomial ring over a field. See [5.2, [8]].

Theorem 1.1.2. If I ⊂ k[x_{1}, …, x_{n}] has no common zero over k̄, then I = (1).

Proof. If I ≠ (1) then there is a maximal ideal 𝔪 such that I ⊂ 𝔪. Let α_{i} be the image of x_{i} in K = k[x_{1}, …, x_{n}]/𝔪. Since K is generated as an algebra over k by α_{1}, …, α_{n} the lemma implies that K is algebraic over k. Furthermore, every element of I vanishes on (α_{1}, …, α_{n}) so (α_{1}, …, α_{n}) is a common zero for I.

Corollary 1.1.3. If I ⊂ k[x_{1}, …, x_{n}], f ∈ k[x_{1}, …, x_{n}], and f vanishes on every common zero of I over k̄, then f^{r} ∈ I for some r.

Proof. The ideal J = I + (1 − yf) ⊂ k[x_{1}, …, x_{n}, y] has no common zeroes over k̄ so 1 ∈ J. Thus there is a relation 1 = ∑ p_{i}h_{i} + q(1 − yf) with h_{i} ∈ I. Making the substitution y ↦ f^{−1} gives the identity

1 = ∑ p_{i}(x_{1}, …, x_{n}, f^{−1})h_{i}

which, upon multiplying through by a sufficiently large power r of f, shows that f^{r} ∈ I.

The Nullstellensatz implies that the coordinate ring of an algebraic set has no nilpotents, and that there is a one-to-one correspondence between algebraic sets and radical ideals.

Example 1.1.4. If f_{1}, …, f_{n} ∈ ℚ[x] have (0, …, 0) as their only common zero over \overline{ℚ}, then for each i, some power of x_{i} lies in (f_{1}, …, f_{n}). For example, if f_{1} = y − x^{2} and f_{2} = y + x^{2} then x^{2} ∈ (f_{1}, f_{2}) and y ∈ (f_{1}, f_{2}).

1.1.2 Nonsingularity

Definition 1.1.5. Let f_{1}, …, f_{t} be a set of generators for the ideal of an affine variety X ⊂ 𝔸_{k}^{n}. Then X is nonsingular at a point p if the rank of the matrix whose i, j entry is ∂f_{i}/∂x_{j}(p) is n − dim X.

1.2 Schemes

Definition 1.2.1. If X = Spec(A) is an affine scheme and f ∈ A, then the distinguished open subset corresponding to f is

D(f) = {𝔭 ∈ X : f ∉ 𝔭}.

Proposition 1.2.2. Any affine scheme X = Spec(A) is quasi-compact.

Proof. Let {U_{i}} be an open cover of X. By refining {U_{i}} we may assume that U_{i} = D(f_{i}) for some f_{i} ∈ A. No prime of A contains all of the f_{i} since otherwise it would be in the complement of the union of the U_{i}. Thus the f_{i} generate the unit ideal, so there exists a_{1}, …, a_{r} such that a_{1}f_{1} + ⋯ + a_{r}f_{r} = 1. It follows that X = U_{1} ∪ ⋯ ∪ U_{r}.

Note that it is not necessary to assume that A is noetherian in the above proposition.

Proposition 1.2.3. If f, g ∈ A then D(f) ∩ D(g) = D(fg).

Proof. A prime ideal contains neither f nor g iff it doesn’t contain their product.

Example 1.2.4. Not every open subset of an affine scheme need be distinguished, although every one can be written as a (possibly infinite) union of distinguished opened subsets. For example, the subset U = 𝔸^{2} − {point} ⊂ 𝔸^{2} is opened but not affine so it is not distinguished. Since U = D(x) ∪ D(y) we also see that the union of distinguished opened subsets need not be distinguished.

1.2.1 Glueing

1.3 Properties of Schemes

Let X be an arbitrary scheme.

1.3.1 Connected, irreducible, reduced, integral, affine

Definition 1.3.1. X is connected if the underlying topological space of X is connected, i.e., cannot be written as a disjoint union of two proper open subsets.

Proposition 1.3.2. An affine scheme X = Spec A is not connected iff A ≅ A_{1} × A_{2} with A_{1} and A_{2} nonzero rings.

Proof. If X is the disjoint union of open subset U_{1} and U_{2} then, because O_{X} is a sheaf, the map

A = Γ(X, O_{X}) ⟶^{res_{U_{1}} ⊕ res_{U_{2}}} O_{X}(U_{1}) ⊕ O_{X}(U_{2})

is an isomorphism. Conversely, if X = Spec A and A = A_{1} × A_{2} then X is the union of disjoint open subsets Spec A_{1} ∪ Spec A_{2} so X is not connected.

Definition 1.3.3. X is integral if for every open subset U ⊂ X, O_{X}(U) is an integral domain.

The property of being integral is a global property not a local property. It is not enough to insist that there exists a covering of X by open sets U such that O_{X}(U)is an integral domain. If for every point 𝔭 ∈ X, the local ring O_{X,𝔭} is an integral domain one still can not conclude that X is integral. Example 1.3.4. Let X be the disjoint union 𝔸^{1} ∪ 𝔸^{1}. Then X is covered by two open subsets whose global sections are integral domains. Also, every local ring of X is a domain. But X is not integral.

Definition 1.3.5. X is irreducible if its topological space can not be written as a union of two proper closed subsets. X is reduced if for every point x ∈ X the local ring O_{X,x} has no nilpotents.

A scheme is integral iff it is reduced and irreducible.

1.3.2 Geometrically reduced and irreducible

Let X be a scheme of finite type over a field k.

Definition 1.3.6. X is geometrically irreducible if X ×_{k} k̄ is irreducible. X is geometrically reduced if X ×_{k} k̄ is reduced.

Proposition 1.3.7. If X is geometrically irreducible (resp., reduced) then X ×_{k}K is irreducible (resp., reduced) for all extension fields K of k.

Example 1.3.8. Let k = 𝔽_{p}(t) and let X = Spec k[x]/x^{p} − t. Then X is of finite type over k and is the spectrum of a field hence integral. But X is not geometrically reduced because

Γ(Xk̄, O_{Xk̄}) = k̄[x]/x^{p} − t

and there is s ∈ k̄ with s ≠ x (modulo x^{p} − t) and s^{p} = t so that (s − x)^{p} = s^{p} − x^{p} = 0 so s − x is nilpotent. Thus X is integral but not geometrically reduced.

Let k = ℚ and let X = Spec ℚ[i]. Then X is integral but

X\overline{ℚ} = Spec \overline{ℚ}[x]/(x^{2} + 1)

which is a union of two points, hence reducible. Thus X is integral but not geometrically irreducible.

1.3.3 Locally noetherian and noetherian

Definition 1.3.9. A scheme X is locally noetherian if there is a covering of X by open affines of the form Spec(A) where A is a noetherian ring. X is noetherian if the covering can be taken to be finite.

Proposition 1.3.10. If X is locally noetherian and Spec(A) ⊂ X is any open affine subset of X then A must be noetherian.

1.3.4 Normal

Definition 1.3.11. X is normal if every local ring of X is an integrally closed domain.

If A is integrally closed then, because localization commutes with integral closure, Spec A is normal.

Example 1.3.12. Spec(ℤ[\sqrt{5}]) is not normal because the local ring at 2 is not integrally closed.

Proposition 1.3.13. A noetherian integral normal scheme is regular in codimension 1. In particular a normal curve is nonsingular.

Proof. A one-dimensional local noetherian ring is regular iff it is integrally closed.

Proposition 1.3.14. Let X be an integral scheme. Then there exists a normal scheme X̃ along with a morphism f : X̃ → X which has the following universal property. If Z is a normal integral scheme then any dominant morphism Z → X factors uniquely through f. If X is of finite type over a field then f is finite.

Proof. This is exercise II.3.8 in [5]

Example 1.3.15. The two dimensional quadratic cone is normal, but singular. The cuspidal cubic y^{2} = x^{3} is not normal because \frac{y}{x} ∉ k[x, y]/(y^{2} − x^{3}) but satisfies the integral equation

(\frac{y}{x})^{2} − x = 0. The nodal cubic y^{2} = x^{2}(x − 1) is not normal because \frac{y}{x} satisfies the integral

equation (\frac{y}{x})^{2} − x + 1 = 0.

Proposition 1.3.16. Suppose X = Proj(A[x_{0}, …, x_{r}]/I) is a closed subscheme of ℙ_{A}^{n}. Then A[x_{0}, …, x_{r}]/I is integrally closed iff X is normal and for every n ≥ 0 the natural map Γ(ℙ^{r}, O_{ℙ^{r}}(n)) → Γ(X, O_{X}(n)) is surjective.

This is exercise II.5.14 in [5].

Example 1.3.17. The rational quartic curve in ℙ^{4} given parametrically by

(x, y, z, w) = (t^{4}, t^{3}u, tu^{3}, u^{4})

is a normal scheme but its coordinate ring k[x, y, z, w]/(xw − yz, y^{3} − x^{2}z, z^{3} − yw^{2}, y^{2}w − xz^{2})

is not integrally closed because (\frac{yw}{z} )^{2} = xw.

1.3.5 Regular

A scheme is regular if all of its local rings are regular local rings.

A variety X over a field k is nonsingular if Xk̄ is regular. If X is nonsingular then X_{ℓ} is regular for all fields k ⊂ ℓ ⊂ k̄.

Example 1.3.18. There are examples of varieties X/k which are regular but such that Xk̄ is not regular. Let k = 𝔽_{2}(t) and let X be the affine curve y^{2} = x(x^{2} − t). Then Xk̄ is defined by y^{2} = x(x − \sqrt{t})^{2} which has a nodal singularity. But, the local ring of X corresponding to the maximal ideal 𝔪 = (y, x^{2} − t) is regular because localizing inverts x so that x^{2} − t = y/x and hence the maximal ideal is generated by y.

If k is a perfect field, there can not be an example of a variety X/k which is regular but whose base extension Xk̄ is not regular. When k is perfect, I think that the sheaf Ω_{X/k} of differentials is locally free and hence its base extension to k̄ is also locally free. (see the proof of Theorem II.8.15 in [5])

Let X be a variety of dimension d over an algebraically closed field. Then the local ring corresponding to a closed point P of X has dimension d and is regular iff P sits in a nonsingular way on X. If R is a local ring of X of dimension less than d, let V be the set of points on X corresponding to the maximal ideals which contain the ideal defining R. Because a localization of a regular local ring is regular, if one of the points in V is nonsingular then R is regular.

Example 1.3.19 (Regular in codimension one). Every one dimensional local ring of the cone x^{2}+y^{2} = z^{2} is regular even though the cone has a singular point. Let R = k[x, y, z]/(x^{2}+y^{2}−z^{2}) and let 𝔭 = (x, y−z). Then R_{𝔭} has dimension one and 𝔭R_{𝔭} is principal since y−z = x^{2}/(y+z) ∈ R_{𝔭}. Although R_{𝔭} is regular, it is also a localization of R_{(x,y,z)} which is not regular.

1.4 Sheaves

Example 1.4.1. This is an example of a presheaf on an affine scheme which is not a sheaf. Let X be an affine scheme and let F be any sheaf on X possessing a nonzero global section. Let F′ be the presheaf

F′(U) = \begin{cases}F(U) if U ≠ X \\ 0 if U = X \end{cases}

If X possesses a covering by proper open subsets then F′ can not be a sheaf because the nontrivial global section of F would, by the sheaf axioms, give rise to a nontrivial global section of F′.

1.4.1 Operations on sheaves

Definition 1.4.2. Let X be a scheme and let F and G be sheaves of O_{X}-modules. The tensor product F ⊗ G is the sheaf associated to the presheaf

U ↦ F(U) ⊗_{O_{X}(U)} G(U).

Example 1.4.3. If X = ℙ_{k}^{n} then O_{X}(1) ⊗ O_{X}(−1) = O_{X} which has nontrivial global sections whereas Γ(X, O_{X}(1)) ⊗ Γ(X, O_{X}(−1)) = 0 since Γ(X, O_{X}(−1)) = 0. Thus for X the presheaf tensor product is not a sheaf. Question: On an affine scheme, is the presheaf tensor product a sheaf? Probably not – a counterexample might be found by considering a non-distinguished opened subset of an affine scheme, for example, the affine plane minus the origin.

1.5 Direct and inverse image functors

Definition 1.5.1. Let f : X → Y be a morphism of schemes and let F be a sheaf on X. Then direct image f_{∗}F is the sheaf U ↦ F(f^{−1}(U)) with the natural restriction maps. The induced map f^{#} : O_{Y} → f_{∗}O_{X} gives f_{∗}F the structure of O_{Y} -module.

Definition 1.5.2. Let f : X → Y be a morphism of schemes and let F be a sheaf of O_{Y} - modules. Then f^{−1}F is the sheaf associated to the presheaf

U ↦ lim_{V ⊃f(U)} F(V)

with the natural restriction maps. f^{−1}F is naturally an f^{−1}O_{Y} -module. The map f^{#} : O_{Y} → f_{∗}O_{X} gives rise to a natural map f^{−1}O_{Y} → O_{X}. Define the inverse image

f^{∗}F = f^{−1}F ⊗_{f^{−1}O_{Y}} O_{X}.

In fact f_{∗} and f^{∗} are adjoint functors so there is a natural isomorphism

Hom_{O_{X}} (f^{∗}G, F) ≅ Hom_{O_{Y}} (G, f_{∗}F).

Proposition 1.5.3. Suppose F and G are sheaves of O_{Y} -modules and f : X → Y. Then f^{∗}(F ⊗ G) = f^{∗}(F) ⊗ f^{∗}(G).

Proof. Ex. II.5.16(e).

1.5.1 Quasi-coherent and coherent sheaves

Definition 1.5.4. Let X = Spec(A) be an affine scheme and let M be an A-module. Then M̃ is the sheaf of O_{X}-modules such that for any f ∈ A, M̃(D(f)) = M_{f}, with the natural restriction maps.

The above definition makes sense because any open subset U ⊂ X is a union of distinguished opened subsets D(f). Indeed, U is of the form {𝔭 ∈ X : 𝔭 ⊅ I} for some ideal I ⊂ A, so U = ∪_{f∈I}D(f).

Proposition 1.5.5. Let X = Spec(A) be an affine scheme and let M and N be A-modules. Then

1. for each 𝔭 ∈ X, ˜(M)_{𝔭} = M_{𝔭},

2. M ↦ M̃ is a fully faithful functor

{ modules over A } ⟶ { sheaves of O_{X}-modules },

3. (M ⊗_{A} N)^{}˜ = M̃ ⊗_{O_{X}} Ñ and (M ⊕ N)^{}˜ = M̃ ⊕ Ñ,

4. f_{∗}(Ñ) = _{ Ã}N and f^{∗}(M̃) = (M ⊗_{A} B)^{}˜ .

Note that˜ is not an equivalence of categories. For example, if X = Spec(A) is irreducible and i : U → X makes U into a proper open subscheme, then i_{!}O_{U}, the sheaf obtained by extending O_{U} by zero outside of U, is a sheaf of O_{X}-modules. If V is a nonempty affine open subset of X which does not contain U, then i_{!}O_{U}|_{V} has no nonzero global sections but is not the zero sheaf. Thus i_{!}O_{U} is not M̃ for any A-module M.

Definition 1.5.6. Let X be an arbitrary scheme and F a sheaf of O_{X}-modules. Then F is a quasi-coherent sheaf if there is a covering {U_{i} = Spec(A_{i})} of X by open affines such that for each i, F|_{U_{i}} ≅ M̃_{i} for some A_{i}-module M_{i}. If each M_{i} is a finitely generated A_{i}-module then F is coherent.

Proposition 1.5.7. Let X be a scheme, then the category of sheaves on X is closed under taking kernels, cokernels, images, and extensions.

Proposition 1.5.8. Let f : X → Y be a morphism of schemes. Then f induces a map

f^{∗} : { quasi-coherent O_{Y} -modules } ⟶ { quasi-coherent O_{X}-modules }.

If X is noetherian, then f also induces a map in the other direction

f_{∗} : { quasi-coherent O_{X}-modules } ⟶ { quasi-coherent O_{Y} -modules }.

The above two propositions admit generalizations. Example 1.5.9. Let f : X → Y be a morphism of schemes. If F is coherent on X then f_{∗}F need not be coherent on Y. For example, let f : 𝔸^{1} → Spec k, then F = O_{𝔸^{1}} is coherent but f_{∗}F is not coherent since k[x] is not finitely generated over k. Note that f is not finite.

Theorem 1.5.10. If f : X → Y is finite or proper and F is a coherent sheaf on X then f_{∗}F is a coherent sheaf on Y.

Proposition 1.5.11. Let X be a scheme. Then there is a bijection

{ closed subschemes of X } → { quasi-coherent sheaves of O_{X}-ideals }
(Y ↪ X) ↦ I = ker(f^{#} : O_{X} → f_{∗}O_{Y})

Quasicoherent sheaves on Proj(S)

Definition 1.5.12. Let S be a graded ring which defines the scheme X = Proj(S) and let M be a graded S-module. Define M̃ to be the sheaf of O_{X}-modules such that M̃(D_{+}(f)) = { \atop M_{(f)}}. The sheaf M̃ is quasicoherent.

Definition 1.5.13. If X = Proj(S) and n ∈ ℤ denote by O_{X}(n) the sheaf { \atop S(n)}. Furthermore, for any sheaf F of O_{X}-modules, define the twist of F by n to be F(n) = F ⊗ O_{X}(n).

If S is a graded ring we will say that S is generated in degree 1 if S is generated by S_{1} as an S_{0}-algebra.

Proposition 1.5.14. Suppose S is a graded ring generated in degree 1 and let X = Proj(S). Then

1. if F is an invertible sheaf then so is F(n) for any integer n,

2. { \atop M(n)} = M̃(n), in particular, O_{X}(n) ⊗ O_{X}(m) = O_{X}(n + m),

3. if T is another graded ring, generated in degree 1, φ : S → T is a degree preserving homomorphism, U = {𝔭 : S_{+} ⊄ φ^{−1}(𝔭)} ⊂ Y, and f : U → X the induced morphism, then for any grades S-module M, f^{∗}(M̃) = (M ⊗_{S} T)^{}˜ |_{U} and for any graded T -module N, f_{∗}(Ñ|_{U}) = Ñ.

Definition 1.5.15. Let X = Proj(S) and let F be a sheaf of O_{X}-modules. The additive group

Γ_{∗}(F) = ⊕_{n∈ℤ}Γ(X, F(n))

has a natural structure of graded S-module.

Proposition 1.5.16. Suppose X = Proj(S) with S finitely generated in degree 1, and that F is a quasi-coherent sheaf on X. Then there is a natural isomorphism { \atop Γ_{∗}(F)} → F.

Theorem 1.5.17. Let A be a finitely generated algebra over a field k, let X be a projective scheme over A, and let F be a coherent O_{X}-module. Then H^{0}(X, F) is a finitely generated A-module.

1.5.2 Locally free sheaves

Definition 1.5.18. A sheaf F of O_{X}-modules is locally free if for every point x ∈ X the stalk F_{x} is a free O_{X,x}-module.

If F is locally free then for every point x ∈ X there is an open neighborhood U of x such that F|_{U} is a free O_{X}|_{U}-module. If X can be covered by open sets U such that F|_{U} is a free O_{X}|_{U}-module then F is locally free.

Proposition 1.5.19 (Projection Formula). Let f : X → Y be a morphism of schemes, let F be a sheaf of O_{X}-modules, and let E be a locally free sheaf of O_{Y} -modules. Then

f_{∗}(F ⊗_{O_{X}} f^{∗}E) ≅ f_{∗}(F) ⊗_{O_{Y}} E.

Example 1.5.20. The push forward of a locally free sheaf need not be locally free. For example, let R be a ring, let I be any nontrivial ideal, and let f : Spec R/I → Spec R be the natural closed immersion. The the structure sheaf on Spec R/I is locally free but its pushforward to Spec R is not free.

1.6 Properties of Morphisms

In this section X and Y denote schemes.

1.6.1 Quasi-compact

Definition 1.6.1. f : X → Y is quasi-compact if for every open affine V ⊂ Y, f^{−1}(V) is quasi-compact.

1.6.2 Finite type and locally of finite type

Definition 1.6.2. f : X → Y is locally of finite type if for every open affine V = Spec B ⊂ Y, f^{−1}(V) can be covered by open affines U_{i} = Spec A_{i}, where each A_{i} is a finitely generated B-algebra. The morphism f is of finite type if in addition f^{−1}(V) can be covered by a finite number the U_{i}.

In fact, to check that a morphism is of finite type it sufficies to exhibit a single covering of Y by such open affines V.

Proposition 1.6.3. 1. Closed immersions and quasi-compact open immersions are of finite type.

2. If f : X → Y and g : Y → Z are of finite type then g ∘ f : X → Z is of finite type.

3. The property of being of finite type is closed under base extension. Thus if X → Y is of finite type and Y′ → Y is a morphism then the induced morphism X ×_{Y} Y′ → Y′ is of finite type.

4. If X and Y are of finite type over a base scheme S then X ×_{S} Y is of finite type over S.

5. If X is of finite type over a field then the set of closed points of X is dense.

1.6.3 Finite and Quasi-finite

Definition 1.6.4. f : X → Y is a finite morphism if for every open affine V = Spec B ⊂ Y, f^{−1}(V) is affine, equal to Spec A, where A is a B-algebra which is finitely generated as a B-module.

Definition 1.6.5. f : X → Y is quasi-finite if for every point y ∈ Y the inverse image f^{−1}(y) is a finite set.

Proposition 1.6.6. A finite morphism is proper and quasi-finite.

Proof. See [5] exercises II.3.5 and II.4.1.

Theorem 1.6.7. A projective and quasi-finite morphism is finite.

Proof. See [5] exercise III.11.2.

Proposition 1.6.8. A closed immersion is finite.

Example 1.6.9. If O is an order in the ring of integers of a number field then O is finitely generated as a ℤ-module so Spec O is finite over Spec ℤ; hence Spec O is proper over Spec ℤ.

Projection from the hyperbola xy = 1 to the x-axis is quasi-finite but not finite.

1.6.4 Affine

1.6.5 Immersions

Definition 1.6.10. An open subscheme of a scheme X is an open subset U ⊂ X with structure sheaf O_{X}|_{U}. A morphism f : Y → X is an open immersion if f(Y) is open and f induces an isomorphism of schemes Y → f(Y). (where f(Y) is given the scheme structure in which its structure sheaf is O_{X}|_{f(Y)}.)

Definition 1.6.11. A morphism f : Y → X is a closed immersion if it induces a homeomorphism from sp(Y) to a closed subset of sp(X) and if furthermore f^{#} : O_{X} → f_{∗}O_{Y} is surjective.

Note that f^{#} is not a morphism of sheaves of O_{X}-modules but is only a morphism of sheaves and that f^{#} is surjective iff f^{#} is surjective on stalks.

A scheme X over Y with structure morphism f : X → Y is projective if, for some n, it factors as

X ⟶^{i} ℙ_{Y}^{n} = ℙ_{Z}^{n} ×_{ℤ} Y ⟶^{π_{Y}} Y

with i a closed immersion.

Example 1.6.12 (Parabola). The map f : 𝔸^{1} → 𝔸^{2} given by t ↦ (t, t^{2}) is a closed immersion. The image is the closed subset defined by y = x^{2} and the induced map on sheaves k[x, y] → k[t] is given by x ↦ t, y ↦ t^{2}.

Example 1.6.13 (Not surjective on stalks). Let Y = Spec(k[ε]/ε^{2}) and X = Spec(k[ε]/ε^{3}). Define a morphism f : Y → X by

f^{#} : k[ε]/ε^{3} → k[ε]/ε^{2} : ε ↦ ε^{2}.

Then sp(X) and sp(Y) are both one point spaces hence homeomorphic under f, but f^{#} is not surjective.

Proposition 1.6.14. Let I be an ideal of a ring A. The morphism Spec(A/I) → Spec(A) induced by the natural map A → A/I is a closed immersion.

1.6.6 Separated

Definition 1.6.15. A morphism X → S is separated if the diagonal morphism X → X ×_{S} X is a closed immersion.

Proposition 1.6.16. Any morphism between affine schemes is separated.

Proof. If f : Spec(A) → Spec(B) then Δ corresponds to A ⊗_{B} A → A which is a surjective morphism of rings hence f is a closed immersion.

Example 1.6.17. The line with a doubled origin, denoted − : −, is not separated over ℤ.

Theorem 1.6.18 (Valuative criterion for separatedness). A morphism f : X → S is separated iff the following condition is satisfied. Whenever R is a valuation ring with field of fractions K and there is a commutative diagram

Spec(K) ⟶^{i} Spec(R)
↓  ↓
X  ⟶^{f}  S

then there is at most one morphism Spec(R) → X making the diagram commute. (Here i is induced by the inclusion R ↪ K.)

Example 1.6.19. The sequence

{•} → − : − → ℤ

shows that X → Y → Z can be separated even though Y → Z is not separated. (By the corollary X → Y must be separated.)

Corollary 1.6.20. Assume all schemes considered below are noetherian.

1. Open and closed immersions are separated.

2. A composition of two separated morphisms is separated.

3. If X → S is separated and T/S is a scheme over S then X × T → T is separated, thus separated morphisms are stable under base extension.

4. The product of proper morphisms is proper.

5. If f : X → Y are g : Y → Z morphisms and g ∘ f is separated, then X ⟶^{f} Y is separated.

Proposition 1.6.21. Consider schemes over a fixed base scheme S. If X contains a closed (or open) subscheme which is not separated, then X is not separated.

Proof. Suppose f : Y → X is a closed (or open) immersion and that Y is not separated. By the corollary f is separated. Thus if X → S is separated then Y → X → S is separated, a contradiction.

Example 1.6.22. By the proposition the plane with a doubled x-axis is not separated over ℤ.

Proposition 1.6.23. If X is separated over an affine scheme S then the interesection of any two affine subsets of X is again affine.

Example 1.6.24. Let X be the affine plane with a doubled origin. Let A_{1} be the affine subset consisting of the affine plane with exactly one of the origins, and let A_{2} be the affine plane with only the other origin (X can be obtained by glueing A_{1} and A_{2}.) Then A_{1} ∩ A_{2} is the affine plane minus the origin which is not affine. This does not contradict the proposition because X is not separated over any affine scheme.

1.6.7 Proper

f : X → Y is universally closed if for all Y′ → Y the induced morphism f′ : X ×_{Y} Y′ → Y′ is a closed map on the underlying topological spaces.

Definition 1.6.25. f : X → Y is proper if it is separated, of finite type, and universally closed.

Theorem 1.6.26 (Valuative criterion for properness). A morphism f : X → S is proper iff the following condition is satisfied. Whenever R is a valuation ring with field of fractions K and there is a commutative diagram

Spec(K) ⟶^{i} Spec(R)
↓  ↓
X  ⟶^{f}  S

then there exists a unique morphism Spec(R) → X making the diagram commute. (Here i is induced by the inclusion R ↪ K.)

Corollary 1.6.27. Assume all schemes considered below are noetherian.

1. Closed immersions are proper.

2. The composition of proper morphisms is proper.

3. Proper morphisms are stable under base extension.

4. The product of proper morphisms is proper.

5. If f : X → Y and g : Y → Z are morphisms and g ∘ f is proper and g is separated, then f is proper.

Proposition 1.6.28. A proper morphism between affine varieties is finite.

1.6.8 Projective and quasi-projective

Definition 1.6.29. f : X → Y is projective if there is a closed immersion g : X → ℙ_{Y}^{n} = ℙ_{ℤ}^{n} ×_{ℤ} Y such that f = π_{Y} ∘g. f is quasi-projective if there is a scheme X′, an open immersion i : X → X′, and a projective morphism h : X′ → Y such that f = h ∘ i.

Thus X is projective over S if there is a closed immersion of X into some projective space over S which preserves the given structure of S-scheme.

Theorem 1.6.30. A projective morphism of noetherian schemes is proper. A quasi-projective morphism of noetherian schemes is separated and of finite type.

This is proved by showing that ℙ_{ℤ}^{n} is proper over ℤ and then using the fact that compositions and base extensions of proper morphisms are proper when the schemes involved are noetherian.

Proposition 1.6.31. A composition of projective morphisms is projective.

Theorem 1.6.32 (Chow’s Lemma). Let f : X → S be a proper morphism to a noetherian scheme S. Then there exists a scheme X′ and a morphism g : X′ → X such that f ∘ g is projective and there is an open dense subset U ⊂ X such that g : g^{−1}(U) → U is an isomorphism of schemes.

1.6.9 Flat

Definition 1.6.33. A morphism f : X → Y of schemes is flat if for every point x ∈ X the local ring O_{x,X} is a flat O_{f(x),Y} module via the natural map f^{#} : O_{f(x),Y} → O_{x,X}. A sheaf F of O_{X}-modules is flat over Y if for whenever x ∈ X then F_{x} is a flat O_{f(x),Y} -module.

Theorem 1.6.34. 1. Open immersions are flat.

2. Let f : X → Y be a morphism of schemes and let F be a sheaf of O_{X}-modules which is flat over Y. Let g : Y′ → Y be a morphism of schemes. Then the pullback F′ of F to X′ = X ×_{Y} Y′ is flat over Y′.

3. Let X be a noetherian scheme, and F a coherent O_{X}-module. Then F is flat over X iff it is locally free.

Theorem 1.6.35 (Cohomology commutes with flat base extension.). Let f : X → Y be a separated morphism of finite type of neotherian schemes, and let F be a quasi-coherent sheaf of O_{X}-modules. Let u : Y′ → Y be a flat morphism of schemes.

X′ ⟶^{v} X
↓ g  ↓ f
Y′ ⟶^{u} Y

Then for all i ≥ 0 there are natural isomorphisms

u^{∗}R^{i}f_{∗}(F) ≅ R^{i}g_{∗}(v^{∗}F).

Example 1.6.36. For example, if X is a separated noetherian scheme of finite type over a field k and F is a quasi-coherent sheaf on X then for all q ≥ 0,

H^{q}(X, F) ⊗_{k} k̄ ≅ H^{q}(Xk̄, Fk̄).

If X is an irreducible projective variety and F = O_{X} then we see that

H^{0}(X, O_{X}) ⊗_{k} k̄ ≅ H^{0}(Xk̄, O_{Xk̄}) = k̄,

hence H^{0}(X, O_{X}) = k.

Definition 1.6.37. A point x in a scheme X is an associated point of X if the maximal ideal 𝔪_{x} is an associated prime of 0 in the local ring O_{x,X}.

Theorem 1.6.38. Let f : X → Y be a morphism of schemes, with Y integral and regular of dimension 1. Then f is flat iff every associated point x ∈ X maps to the generic point of Y.

Example 1.6.39. Let Y = Spec ℤ and let X = Spec ℤ/pℤ. Then the morphism f : X → Y induced by the natural map ℤ → ℤ/pℤ is not flat because the image of the associated point (0) ∈ X is (p) ∈ Y which is not the generic point of Y. Non-flatness can also be seen directly since ℤ/pℤ is not a flat ℤ-module.

Theorem 1.6.40. Let T be an integral noetherian scheme. Let X ⊂ ℙ_{T}^{n} be a closed subscheme. For each point t ∈ T, we consider the Hilbert polynomial P_{t} ∈ ℚ[z] of the fibre X_{t} considered as a closed subscheme of ℙ_{k(t)}^{n}. Then X is flat over T iff the Hilbert polynomial P_{t} is independent of t. If X is flat then the dimension of X_{t}, the degree of X_{t}, and the arithmetic genus of X_{t} are all independent of t.

Example 1.6.41. Let X be a scheme over a field k. Then X is flat over k because by linear algebra a module over a field k is automatically flat.

Let E be an elliptic curve over ℚ. Fix a Weierstrass equation

y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6}

with a_{i} ∈ ℤ. The Weierstrass equation gives rise to a scheme over ℤ

E = Proj(ℤ[x, y, z]/(zy^{2} + a_{1}xyz + a_{3}yz^{2} − x^{3} − a_{2}x^{2}z − a_{4}xz^{2} + a_{6}z^{3})).

In general a scheme over Spec ℤ is flat iff all of its local rings are torsion free as abelian groups. This is true for E as the following proposition shows.

Proposition 1.6.42. Let f ∈ R = ℤ[x_{1}, …, x_{n}] be a polynomial. Then f ∉ pR for all prime numbers p iff R/(f) is torsion free as an additive abelian group.

Proof. If R/(f) has torsion then it has prime torsion so there is a prime p and an element g ∈ R, g ∉ (f) such that pg ∈ (f). Then there is h ∈ R such that pg = hf so hf ∈ (p). Since (p) is prime and f ∉ (p), it follows that h ∈ (p) so g = \frac{h}{p}f ∈ (f), a contradiction.

If f ∈ pR for some prime p then f = gp for some g ∈ R. Thus g ∉ (f) but pg = f ∈ (f) so g gives rise to a nonzero p-torsion element of R/(f).

Example 1.6.43. The proposition shows that the scheme X = Spec(ℤ[x, y]/(2x^{2} + 2y^{2} − 2)) is not flat over Spec ℤ. Nevertheless, Spec(ℤ[\frac{1}{2}])[x, y]/(2x^{2} + 2y^{2} − 2) is flat over Spec ℤ[\frac{1}{2}].

Example 1.6.44. Suppose K/L is an extension of number fields, then the map Spec O_{L} → Spec O_{K} between their integer rings is flat because O_{K} is a flat O_{L} module. This is because O_{L} is locally a PID and O_{K} is locally torsion free.

1.6.10 Smooth

1.6.11 Étale

1.6.12 Unramified

1.7 Divisors

1.7.1 Weil divisors

Weil divisors are defined for schemes which satisfy the following condition:

(∗) X is a noetherian integral separated scheme which is regular in codimension one.

Definition 1.7.1. A prime divisor on X is a closed integral subscheme Y of codimension one. A Weil divisor is an element of the free abelian group Div X generated by the prime divisors. A prime divisor is effective if the coeffients in the sum are all ≥ 0. If Y is a prime divisor with generic point η then the local ring O_{η,X} is a discrete valuation ring with valuation v_{Y} and quotient field K = K(X). If f ∈ K^{∗} then v_{Y} (f) is an integer n. If n > 0 then f has a zero of order n at Y, and if n < 0 then f has a pole of order −n at Y.

Proposition 1.7.2. If X satisfies (∗) and f ∈ K^{∗} then v_{Y} (f) = 0 for all but finitely many prime divisors Y.

Example 1.7.3. Let \overline{ℤ} denoted the integral closure of ℤ in a fixed algebraic closure \overline{ℚ} of ℚ, and let X = Spec(\overline{ℤ}). Then X is integral and separated but not noetherian. The proposition does not hold for X. (Of course the proposition doesn’t make sense because the local rings of X need not be discrete valuation rings, but we can take the condition v_{Y} (f) = 0 to mean f is a unit in the local ring corresponding to Y.)

We know (example 14.6.1) that every nonzero prime ideal of \overline{ℤ} is maximal so \overline{ℤ} has dimension one and every nonzero prime has height 1, i.e., the closed subschemes Spec \overline{ℤ}/𝔭 of Spec \overline{ℤ} all have codimension one. Suppose O is the ring of integers of a number field and let 𝔭 be a prime of O. By the going-up theorem there is a prime 𝔓 of \overline{ℤ} such that 𝔓 ∩ O = 𝔭. Thus to show that there are infinitely many prime ideals of \overline{ℤ} which contain a fixed prime p ∈ ℤ it suffices to note that there for arbitrarily large integers n there are number fields K such that the set of primes of K lying over p has cardinality n. Indeed, given n distinct primes of K lying over p we obtain n distinct primes of \overline{ℤ} which contain p. Since such number fields exist (see the following paragraph), the number of primes of \overline{ℤ} containing p can not be finite.

Why do such number fields exist? Fix a prime p ∈ ℤ with p ≠ 2. By Dirichlet’s theorem there are infinitely many primes q_{i} congruent to one modulo p, thus infinitely many quadratic extensions ℚ(\sqrt{q_{i}}) in which p splits completely. The result now follows because if a prime splits completely in two linearly disjoint Galois extensions then it must split completely in their compositum, a fact which can be seen by noting that a prime splits completely iff the conjugacy class of its associated Frobenius automorphism is the identiy class, and Frobenius commutes with restriction.

14 CHAPTER 1. ALGEBRAIC GEOMETRY Example 1.7.4. I think that the proposition still holds for a polynomial ring in infinitely many variables over a field k[x_{1}, x_{2}, …] even though such a ring is not noetherian. I also think that the primes of height one are all principal, generated by an irreducible. [Comment: The maps k[x_{1}, …, x_{n}] ↪ k[x_{1}, …, x_{n}, …] have natural splittings whereas the maps O_{K} ↪ \overline{ℤ} do not.]

1.7.2 Cartier divisors 1.7.3 Invertible sheaves 1.7.4 Picard groups

1.8 Morphisms to projective space 1.9 Basic sheaf cohomology 1.10 Algebraic curves

1.10.1 Dictionary between fields and curves

Theorem 1.10.1. Let k be an algebraically closed field. Then the functor which sends a complete nonsingular curve over k to its function field defines an equivalence of categories

\begin{cases}complete nonsingular \\ curves X/k \end{cases} \} ⟶^{X↦K(X)} \begin{cases}functions fields \\ K/k of tr.deg 1 \end{cases}\}. ********* End Unverified *********

Lie Algebras

2.1 Definitions

Let k be an arbitrary field.

Definition 2.1.1. A lie algebra over k is a k-vector space 𝔤 equipped with a bilinear form [,] : 𝔤 × 𝔤 → 𝔤 such that [x, x] = 0 for all x ∈ 𝔤 and

[x, [y, z]] + [y, [z, x]] + [z, [x, y]] = 0.

2.2 Examples

2.2.1 Linear lie algebras

Let k be an arbitrary field and let V be a vector space over k.

Proposition 2.2.1. The vector space 𝔤𝔩(V) = End_{k}(V) with brack [x, y] = xy − yx is a lie algebra.

Proof. [,] is bilinear because

[λx + y, z] = (λx + y)z − z(λx + y) = λxz + yz − λzx − zy = λ[x, z] + [y, z].

For any x ∈ 𝔤𝔩(V) we have [x, x] = xx − xx = 0. If x, y, z ∈ 𝔤𝔩(V) then

[x, [y, z]] + [y, [z, x]] + [z, [x, y]] = xyz − xzy − yzx + zyx + yzx − yxz
−zxy + xzy + zxy − zyx − xyz + yxz = 0.

Definition 2.2.2. A linear lie algebra is a lie algebra isomorphic to a subalgebra of 𝔤𝔩(V) for some finite dimensional vector space V.

Theorem 2.2.3 (Ado, Iwasawa). Every finite dimensional lie algebra is linear.

Proof. Ado proves this in the characteristic 0 case. The characteristic p case is due to Iwasawa. See chapter VI of Jacobson.

Let V be a finite dimensional vector space over k, and fix a basis v_{1}, …, v_{n} for V. Let e_{ij} be the endomorphism which sends v_{j} to v_{i} and sends v_{k} to 0 for k ≠ j. Then e_{ij}e_{kl} = δ_{jk}e_{il} and so

[e_{ij}, e_{kl}] = e_{ij}e_{kl} − e_{kl}e_{ij} = δ_{jk}e_{il} − δ_{li}e_{kj}.

2.2.2 Classical lie algebras

Type A_{n} Let V be a vector space of dimension n + 1. The special linear lie algebra is

𝔰𝔩(V) = {x ∈ 𝔤𝔩(V) : Tr(x) = 0}.

The dimension of 𝔰𝔩(V) is (n + 1)^{2} − 1.

Type B_{n} Let V be a vector space of dimension 2n + 1 and let (,) be the bilinear form defined by the matrix

B = (\begin{matrix}1 & 0 & 0 \\ 0 & 0 & I_{n} \\ 0 & I_{n} & 0 \end{matrix})

so (v, w) = v^{t}Bw. The odd orthogonal lie algebra is

𝔬(V) = {x ∈ 𝔤𝔩(V) : (xv, w) + (v, xw) = 0 for all v, w ∈ V }.

The dimension of 𝔬(V) is 2n^{2} + n.

Type C_{n} Let V be a vector space of dimension 2n and let (,) be the bilinear form defined by the matrix

(\begin{matrix}0 & I_{n} \\ −I_{n} & 0 \end{matrix}) .

The symplectic lie algebra is

𝔰𝔭(V) = {x ∈ 𝔤𝔩(V) : (xv, w) + (v, xw) = 0 for all v, w ∈ V }.

The dimension of 𝔰𝔭(V) is 2n^{2} + n.

Type D_{n} Let V be a vector space of dimension 2n and let (,) be the bilinear form defined by the matrix

(\begin{matrix}0 & I_{n} \\ I_{n} & 0 \end{matrix}) .

Then the even orthogonal lie algebra is

ø(V) = {x ∈ 𝔤𝔩(V) : (xv, w) + (v, xw) = 0 for all v, w ∈ V }.

The dimension of 𝔰𝔭(V) is 2n^{2} − n.

2.2.3 Other linear lie algebras

Fix a basis for a finite dimensional vector space V. Then the set 𝔱(V) ⊂ 𝔤𝔩(V) of upper triangular matrices is a lie algebra. So is the space 𝔫(V) of upper triangular matrices and the space 𝔡(V) of diagonal matrices.

2.2.4 Derivations

Let A be a k-algebra, that is a vector space over k equipped with an arbitrary bilinear form (,) : A × A → A. Then the space

Der(A) = {δ ∈ End_{k}(A) : δ((x, y)) = (δ(x), y) + (x, δ(y))}

is a lie algebra called the lie algebra of derivations of A. When A = 𝔤 is a lie algebra, there is a map ad : 𝔤 → Der(𝔤) which sends x to the derivation y ↦ [x, y]. The map ad is called the adjoint representation of 𝔤 and plays a very important roll in the theory.

2.2.5 Basic Structure

Definition 2.2.4. An ideal 𝔞 in a lie algebra 𝔤 is a subalgebra such that [𝔤, 𝔞] ⊂ 𝔞. The quotient 𝔤/𝔞 is again a lie algebra. The center of a lie algebra 𝔤 is Z(𝔤) = {x ∈ 𝔤 : [𝔤, x] = 0}.

There is an exact sequence of lie algebras

0 → Z(𝔤) → 𝔤 ⟶^{ad} ad(𝔤) → 0

where ad(𝔤) is a subalgebra of Der(𝔤). Example 2.2.5. Let 𝔤 = 𝔤𝔩(n) and 𝔞 = 𝔱(n) ⊂ 𝔤. The subalgebra 𝔞 is not in general an ideal of 𝔤. For example, take n = 2, x = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ), and y = ( \begin{matrix}1 & 0 \\ 1 & 0 \end{matrix} ). Then

[x, y] = xy − yx = ( \begin{matrix}−1 & −1 \\ 1 & 1 \end{matrix} ) ∉ 𝔞.

Next let 𝔡 ⊂ 𝔤 be the one-dimensional abelian subalgebra of diagonal matrices. Then 𝔡 = Z(𝔤) so 𝔡 is an ideal of 𝔤. What is 𝔤/𝔡? If char k ∤ n then the natural morphism of lie algebras 𝔰𝔩(n) → 𝔤/𝔡 is injective and so it must be an isomorphism because 𝔰𝔩(n) and 𝔤/𝔡 have the same dimension. When char k | n the lie algebras 𝔰𝔩(n) and 𝔤/𝔡 have the same dimension but the map 𝔰𝔩(n) → 𝔤/𝔡 is no longer injective. (Question. Is 𝔤/𝔡 isomorphic to 𝔰𝔩(n) when char k | n?)

Definition 2.2.6. The lie algebra 𝔤 is simple if it is nonabelian and has no proper ideals except 0 and 𝔤.

Example 2.2.7. The special linear lie algebra 𝔰𝔩(n) over a field of characteristic 0 is simple. It is easy to check this directly when n = 2 by picking the basis

x = (\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix}) ,  y = (\begin{matrix}0 & 0 \\ 1 & 0 \end{matrix}) ,  h = (\begin{matrix}1 & 0 \\ 0 & −1 \end{matrix}) .

We have [x, y] = h, [h, x] = 2x, [h, y] = −2y. If 𝔞 ⊂ 𝔰𝔩(2) is an ideal let ax + by + ch be a nonzero element of 𝔞 and apply ad(x) twice, then ad(y) twice to show that 𝔞 = 𝔰𝔩(2).

Definition 2.2.8. The normalizer of a subalgebra 𝔥 ⊂ 𝔤 is

N_{𝔤}(𝔥) = {x ∈ 𝔤 : [x, 𝔥] ⊂ 𝔥}.

Thus N_{𝔤}(𝔥) is the largest subalgebra of 𝔤 in which 𝔥 is an ideal. The centralizer of 𝔥 is

C_{𝔤}(𝔥) = {x ∈ 𝔤 : [x, 𝔥] = 0}.

Example 2.2.9. Let 𝔤 = 𝔤𝔩(n) and let 𝔥 = 𝔰(n) be the n-dimensional abelian subalgebra of scalar matrices. Then N_{𝔤}(𝔥) = C_{𝔤}(𝔥) = 𝔥. To see this note that if x = ∑ α_{ij}e_{ij} ∈ 𝔤 then

[e_{ii}, x] = α_{i1}e_{i1} + ⋯ + α_{in}e_{in} − (α_{1i}e_{1i} + ⋯ + α_{ni}e_{ni})

so if [e_{ii}, x] ∈ 𝔥, for all i, then x is diagonal, so N_{𝔤}(𝔥) ⊂ 𝔥. Since 𝔥 is abelian,

𝔥 ⊂ C_{𝔤}(𝔥) ⊂ N_{𝔤}(𝔥) ⊂ 𝔥.

Definition 2.2.10. A representation of a lie algebra 𝔤 on a vector space V is a homomorphism of lie algebras ρ : 𝔤 → 𝔤𝔩(V). If ρ is injective then ρ is called faithful.

Definition 2.2.11. An automorphism of a lie algebra 𝔤 is a lie algebra isomorphism φ : 𝔤 → 𝔤. The group of automorphisms of 𝔤 is denoted Aut(𝔤).

If 𝔤 ⊂ 𝔤𝔩(V) then every element of GL(V) gives rise via conjugation to an element of Aut(𝔤). The exponential map

exp(δ) = 1 + δ + \frac{δ^{2}}{2!} + \frac{δ^{3}}{3!} + ⋯

induces a map Der(𝔤) → Aut(𝔤).

2.2.6 Solvability

Definition 2.2.12. The derived series is defined as follows.

𝔤^{(0)} = 𝔤, …, 𝔤^{(i+1)} = [𝔤^{(i)}, 𝔤^{(i)}]

The lie algebra 𝔤 is solvable if 𝔤^{(i)} = 0 for some i ≥ 0.

Example 2.2.13. A typical example of a solvable lie algebra is 𝔱(n). If 𝔤 is finite dimensional then the derived series stabilizes after a finite number of steps. An example of an infinite dimensional lie algebra for which the derived series does not stabilize is 𝔤 = ⨁_{n=1}^{∞} 𝔱(n).

Proposition 2.2.14. Suppose

0 → 𝔥 → 𝔤 → 𝔯 → 0

is an exact sequence of lie algebras. Then 𝔤 is solvable iff both 𝔥 and 𝔯 are solvable.

Corollary 2.2.15. If 𝔞, 𝔟 ⊂ 𝔤 are solvable ideals then 𝔞 + 𝔟 is a solvable ideal.

Proof. The sequence

0 → 𝔟 → 𝔞 + 𝔟 → (𝔞 + 𝔟)/𝔟 ≅ 𝔞/(𝔞 ∩ 𝔟) → 0

is exact. Since 𝔟 is solvable and 𝔞/(𝔞 ∩ 𝔟) is solvable (being the quotient of a solvable ideal) it follows that 𝔞 + 𝔟 is solvable.

Definition 2.2.16. The radical \sqrt{𝔤} of 𝔤 is the unique maximal solvable ideal of 𝔤. A lie algebra 𝔤 is semisimple if \sqrt{𝔤} = 0.

Example 2.2.17. Let n be a positive integer and let k be a field with char k ∤ n. The radical of 𝔤𝔩(n) is the ideal of scalar matrices 𝔰 = \sqrt{𝔤𝔩(n)}. Note that 𝔰 is an ideal since it is the center of 𝔤𝔩(n). The quotient 𝔤𝔩(n)/𝔰 is isomorphic to 𝔰𝔩(n) and 𝔰𝔩(n) is simple.

Example 2.2.18. A simple lie algebra is semisimple since it is not solvable and it has no nonzero proper ideals. But a semisimple lie algebra need not be simple. For example, let 𝔤 = 𝔰𝔩(2)⊕𝔰𝔩(2). Then neither of the two nontrivial ideals of 𝔤 are solvable (each being isomorphic to 𝔰𝔩(2)) so \sqrt{𝔤} = 0. Thus 𝔤 is semisimple even though 𝔤 is not simple.

Theorem 2.2.19. Suppose 𝔤 is a finite dimensional semisimple lie algebra over a field of characteristic zero. Then there exists simple ideals 𝔞_{1}, …, 𝔞_{n} ⊂ 𝔤 such that 𝔤 = 𝔞_{1} ⊕ ⋯ ⊕ 𝔞_{n}.

The proof uses Cartan’s criterion and nondegeneracy of the killing form.

2.2.7 Nilpotence

Definition 2.2.20. Let 𝔤 be a lie algebra. Its descending central series is defined as follows.

𝔤^{0} = 𝔤, …, 𝔤^{i+1} = [𝔤, 𝔤^{i}]

We say 𝔤 is nilpotent if 𝔤^{i} = 0 for some i.

Example 2.2.21. A nilpotent lie algebra is solvable because 𝔤^{(i)} ⊂ 𝔤^{i}, but the converse is not true. For example, 𝔱(n) is solvable but not nilpotent when n ≥ 2. The lie algebra 𝔫(n) of strictly upper triangular matrices is a typical example of a nilpotent lie algebra. The abelian lie algebra 𝔡(n) of diagonal matrices is also nilpotent, as is any abelian lie algebra. Thus there is an exact sequence

0 → 𝔡(n) → 𝔱(n) → 𝔫(n) → 0

in which the first and last terms are nilpotent but the center term is not.

Proposition 2.2.22. Suppose

0 → 𝔥 → 𝔤 → 𝔯 → 0

is an exact sequence of lie algebras. If 𝔤 is nilpotent then 𝔥 and 𝔯 must also be nilpotent.

Proposition 2.2.23. If 𝔤/Z(𝔤) is nilpotent then 𝔤 is nilpotent. If 𝔤 ≠ 0 is nilpotent then Z(𝔤) ≠ 0.

Engel’s theorem provides a criterion for nilpotence in terms of the image of the adjoint representation. To prove Engel’s theorem we first prove a lemma.

Lemma 2.2.24. Let V be finite dimensional vector space and suppose 𝔤 ⊂ 𝔤𝔩(V) is a lie subalgebra consisting of nilpotent endomorphisms. Then there exists x ∈ V − {0} such that 𝔤.x = 0.

Proof. We proceed by induction on dim 𝔤, the case dim 𝔤 = 0, 1 being trivial.

1) If 𝔥 ⊂ 𝔤 is a proper nonzero subalgebra then 𝔥 acts on 𝔤/𝔥 so by induction there exists y ∈ 𝔤 such that y + 𝔥 is killed by 𝔥. To apply the inductive hypothesis we are viewing 𝔥 as acting on 𝔤/𝔥 through the adjoint representation and applying the inductive hypothesis to the image of 𝔥 in Der(𝔤/𝔥) ⊂ 𝔤𝔩(𝔤/𝔥). Notice that y ∈ N_{𝔤}(𝔥).

2) Choose a maximal subalgebra 𝔥 of 𝔤. Then N_{𝔤}(𝔥) is a subalgebra which, by part 1, properly contains 𝔥. Hence N_{𝔤}(𝔥) = 𝔤 so that 𝔥 is an ideal in 𝔤. The quotient 𝔤/𝔥 thus makes sense and contains no proper nonzero subalgebras, hence it must have dimension 1.

3) Letting 𝔥 be as in part 2, there exists z ∈ 𝔤 − 𝔥 such that 𝔤 = 𝔥 + k.z. Let

W = {x ∈ V : 𝔥.x = 0} ⊂ V.

Since dim 𝔥 < dim 𝔤 the induction hypothesis implies that W ≠ 0. Since 𝔥 is an ideal, 𝔤 leaves W invariant. In particular z leaves W invariant so, since z is a nilpotent endomorphism, there must be some x ∈ W − {0} such that z.x = 0. Thus there is some x such that 𝔤 annihilates x, as claimed.

Theorem 2.2.25 (Engel). A lie algebra 𝔤 is nilpotent iff the image of ad : 𝔤 → Der(𝔤) consists of nilpotent endomorphisms.

Proof. (⇒) This follows immediately from the definitions. (⇐) By the lemma there exists z ∈ 𝔤 such that [𝔤, z] = 0 so Z(𝔤) ≠ 0 and thus dim 𝔤/Z(𝔤) < dim 𝔤. Since 𝔤/Z(𝔤) has nilpotent image in Der(𝔤/Z(𝔤)) we may apply induction to conclude that 𝔤/Z(𝔤) is nilpotent. It follows that 𝔤 is nilpotent.

Corollary 2.2.26. Suppose 𝔤 ⊂ 𝔤𝔩(V) and that 𝔤 consists of nilpotent endomorphisms. Then there exists a flag stable under 𝔤. In particular, there is a basis of V with respect to which every element of 𝔤 is upper triangular.

Proof. From the lemma we know that there is some x ∈ V such that 𝔤.x = 0. Thus 𝔤 acts on V/k.x. By induction there is a flag in V/k.x stable under the action of 𝔤. Lift 0 to V to obtain the required flag.

2.3 Lie’s theorem

In this section the base field k will be assumed to have characteristic 0.

Theorem 2.3.1 (Lie). Let V be a finite dimensional vector space and let 𝔤 be a solvable lie subalgebra of 𝔤𝔩(V). Then there is a basis of V with respect to which every matrix in 𝔤 is upper triangular.

Corollary 2.3.2. Suppose 𝔤 is a finite dimensional solvable lie algebra. Then there exists an ascending chain of ideals

0 ⊂ 𝔞_{1} ⊂ 𝔞_{2} ⊂ ⋯ ⊂ 𝔞_{n} ⊂ 𝔤

with dim 𝔞_{i} = i.

Proof. The image of 𝔤 in Der 𝔤 is solvable so there is a basis b_{1}, …, b_{n} with respect to which it is upper triangular. Taking 𝔞_{i} to be the subspace spanned by b_{1}, …, b_{i} completes the proof.

2.3.1 Some remarks on Jordan and rational canonical form.

Let A be an n × n matrix over a field k and let V be n dimensional vector space over k. When k is algebraically closed then A is conjugate to a matrix which is a block direct sum of jordan blocks. This is the jordan canonical form of A. A k × k jordan block is the sum of a scalar multiple of the identity with a nilpotent matrix which has ones directly above the diagonal and zeroes elsewhere. The jordan canonical form need not be defined over the same field as A. For example, let A = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ). Then the jordan canonical form of A is ( \begin{matrix}i & 0 \\ 0 & i \end{matrix}).

Theorem 2.3.3 (Jordan). If x ∈ End(V) then x = x_{s} + x_{n} with x_{s} diagonalizable and x_{n} nilpotent. Furthermore, x_{s} = p(x) and x_{n} = q(x) with p and q polynomials without constant term.

When k is not algebraically closed one has the rational canonical form. The can be easily written down once one knows the elementary divisors of A. These are a list {p_{1}^{e_{1}}, …, p_{k}^{e_{k}}} of powers of irreducible polynomials. There is a way to decompose V as a direct sum V_{1} ⊕ ⋯ ⊕ V_{k} such that p_{i}^{e_{i}} is the minimal polynomial of A on V_{i}. Example 2.3.4. Let

A = (\begin{matrix}i & 1 & 1 \\ 0 & −i & 0 \\ 0 & 0 & 1 \end{matrix}) .

The elementary divisors of A are x^{2} + 1 and x − 1. The rational canonical form is thus

(\begin{matrix}1 & 0 & 0 \\ 0 & 0 & −1 \\ 0 & 1 & 0 \end{matrix}) .

The jordan canonical form of the matrix

A = (\begin{matrix}1 & 2 & 7 \\ 0 & 1 & 1 \\ 1 & 0 & 2 \end{matrix})

is diagonal with entries corresponding to the roots of m(x) = (x + 1)(x^{2} − 5x + 3). The rational canonical form is

(\begin{matrix}−1 & 0 & 0 \\ 0 & 0 & −3 \\ 0 & 1 & 5 \end{matrix}) .

Example 2.3.5. Let A/k̄ be an n × n matrix. Let

A = (\begin{matrix}0 & i & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & −i \\ 0 & 0 & 1 & 0 \end{matrix}) .

Then the elementary divisors of A over ℚ(i) are p_{1}(x) = x^{2} − i and p_{2}(x) = x^{2} + i and A is in rational canonical form over ℚ(i). On the other hand x^{4} + 1 is the only possible elementary divisor of A if A is defined over ℚ and the rational canonical form of A over ℚ must then be

(\begin{matrix}0 & 0 & 0 & −1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{matrix}) .

It is possible for a matrix to have a characteristic polynomial which is defined over k but for the matrix to not be conjugate to a matrix over k. For example, let

A = (\begin{matrix}i & 0 & 0 & 0 \\ 0 & i & 0 & 0 \\ 0 & 0 & −i & 1 \\ 0 & 0 & 0 & −i \end{matrix}) .

Then the minimal polynomial is (x − i)(x + i)^{2} ∉ ℚ[x]. The characteristic polynomial is (x − i)^{2}(x + i)^{2} ∈ ℚ[x].

Lemma 2.3.6. If x ∈ 𝔤𝔩(V) and x = x_{s} +x_{n} then ad(x) = ad(x_{s})+ad(x_{n}) is the decomposition of ad(x) into its semisimple and nilpotent parts.

Proof. It suffices to verify, using an explicit computation, that if y ∈ 𝔤𝔩(V) is diagonalizable then ad(y) is diagonalizable.

Lemma 2.3.7. Suppose A is a finite dimensional k-algebra. If x ∈ Der A then x_{s}, x_{n} ∈ Der A.

Proof. This is proved by using the jordan decomposition for x, picking a basis for A and then working out what x_{s} must do and hence verifying that x_{s} is a derivation.

2.4 Cartan’s Criterion

Theorem 2.4.1 (Cartan’s criterion). Let 𝔤 be a lie subalgebra of 𝔤𝔩(V). Then Tr(xy) = 0 for all x ∈ 𝔤 and all y ∈ [𝔤, 𝔤] iff 𝔤 is solvable.

Corollary 2.4.2. Let 𝔤 be a finite dimensional lie algebra. If for all x ∈ 𝔤 and all y ∈ [𝔤, 𝔤], Tr(ad(x) ad(y)) = 0 then 𝔤 is solvable.

Example 2.4.3. Using Cartan’s criterion we verify that the following 3 dimensional lie algebra is solvable. Let 𝔤 be the lie algebra with basis x, y, z, and multiplication [x, y] = z, [x, z] = y, [y, z] = 0. Then [𝔤, 𝔤] is spanned by y and z. To verify Cartan’s criterion we need to check that Tr(ad(a) ad(b)) = 0 for a ∈ {x, y, z} and b ∈ {y, z}. This is easily checked with

ad(x) = (\begin{matrix}0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{matrix}) ,  ad(y) = (\begin{matrix}0 & 0 & 0 \\ 0 & 0 & 0 \\ −1 & 0 & 0 \end{matrix}) ,  ad(z) = (\begin{matrix}0 & 0 & 0 \\ −1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix})

2.5 The Killing form

Given a finite dimensional lie algebra 𝔤 there is a naturally defined symmetric bilinear pairing called the Killing form. In characteristic zero the Killing form is nondegenerate iff 𝔤 is semisimple.

Definition 2.5.1. Let 𝔤 be a finite dimensional lie algebra over a field. The Killing form is the pairing

κ : 𝔤 × 𝔤 → k : (x, y) ↦ Tr(ad(x) ad(y)).

The radical of the killing form is

\sqrt{κ} = \{x ∈ 𝔤 : κ(x, y) = 0 for all y ∈ 𝔤\}.

Lemma 2.5.2. Let 𝔤 be a finite dimensional lie algebra and let 𝔞 be an ideal in 𝔤. Then the Killing form on 𝔞 is the restriction to 𝔞 of the Killing form on 𝔤.

Example 2.5.3. The lemma is false if 𝔞 is replaced by an arbitrary subalgebra of 𝔤. For example, suppose 𝔞 is one dimensional hence abelian. Then the Killing form on 𝔞 is trivial but the restriction of the Killing form on 𝔤 to 𝔞 need not be trivial.

Theorem 2.5.4. Let 𝔤 be a finite dimensional lie algebra over a field of characteristic 0. Then the Killing form κ on 𝔤 is nondegenerate iff 𝔤 is semisimple.

Theorem 2.5.5. Let 𝔤 be a finite dimensional semisimple lie algebra over a field of characteristic 0. Then there exist simple ideals 𝔞_{i} of 𝔤 such that 𝔤 = 𝔞_{1} ⊕ ⋯ ⊕ 𝔞_{n}. The ideals 𝔞_{i} are unique.

Proof. Let 𝔞 be a minimal nonzero ideal of 𝔤. Let 𝔞^{⊥} = {x ∈ 𝔤 : κ(x, 𝔞) = 0}. Then 𝔞 ∩ 𝔞^{⊥} is contained in the radical of κ and is hence 0 because 𝔤 is semisimple. Thus 𝔤 = 𝔞 ⊕ 𝔞^{⊥}. Since 𝔞 is a minimal ideal and κ restricted to 𝔞 is nondegenerate, 𝔞 must be simple. By induction 𝔞^{⊥} is a direct sum of simple ideals (these are also ideals of 𝔤).

Theorem 2.5.6. If 𝔤 is a semisimple lie algebra in characteristic 0, then the adjoint representation ad : 𝔤 → Der 𝔤 is an isomorphism.

Proof. Because κ is nondegenerate, Der 𝔤 splits as a direct sum Der 𝔤 = ad(𝔤) ⊕ 𝔞 for some ideal 𝔞 of Der 𝔤. Thus [𝔞, ad(𝔤)] = 0 so for any δ ∈ 𝔞 and x ∈ 𝔤, 0 = [δ, ad(x)] = ad(δx). Since ad is injective we conclude that δ = 0. Thus 𝔞 = 0, as required.

2.6 Basic facts about representations

Throughout this section let 𝔤 be a lie algebra over a field of characteristic 0.

Definition 2.6.1. A representation of a lie algebra 𝔤 on a vector space V is a lie algebra homomorphism φ : 𝔤 → 𝔤𝔩(V). V is a module over 𝔤 (we denote the action of 𝔤 on V by x.v = φ(x)v.) V is called irreducible if it has precisely two submodules (itself and 0). V is completely reducible if V is a direct sum of irreducible submodules (equivalently, if every submodule splits).

Lemma 2.6.2 (Schur). Let φ : 𝔤 → 𝔤𝔩(V) be irreducible. Then the only endomorphisms of V which commute with φ(𝔤) are the scalars.

Proof. The lemma follows from the simple observation that since V is irreducible any 𝔤-module homomorphism V → V must be either an isomorphism or 0. Suppose ψ is an endomorphism of V. Let λ be an eigenvalue of ψ. Then λ commutes with φ(𝔤) so ψ − λ commutes with φ(𝔤). But ψ − λ has rank < dim(V) so ψ − λ = 0. This shows that ψ is a scalar.

2.6.1 Duality and tensor products

Let 𝔤 be a lie algebra and let V be a 𝔤-module. We give the dual vector space V^{∗} = Hom(V, k) the structure of 𝔤-module as follows: x.f(v) = −f(x.v), for x ∈ 𝔤, f ∈ V^{∗}, and v ∈ V. This gives V^{∗} the structure of 𝔤-module since

[x, y].f(v) = −f([x, y].v) = −f(x.y.v − y.x.v) = x.f(y.v) − y.f(x.v) = −y.x.f(v) + x.y.f(v).

Let V and W be 𝔤-modules. The tensor product V ⊗ W is given the structure of 𝔤-module as follows. If v ∈ V, w ∈ W, and x ∈ 𝔤 then x.(v ⊗ w) = (x.v) ⊗ w + v ⊗ (x.w).

There is a canonical isomorphism V ⊗ V^{∗} ⟶^{≅} End(V) given by sending v ⊗ f to the endomorphism which sends w to f(w)v.

2.6.2 Casimir element

Given a faithful representation φ : 𝔤 → 𝔤𝔩(V) of a semisimple lie algebra 𝔤, and a basis for 𝔤, one associates a particular element 𝔠_{φ} ∈ End(V) called the Casimir element of φ. When the representation V is irreducible the Casimir element relates the dimensions of V and 𝔤.

Fix a faithful representation φ : 𝔤 → 𝔤𝔩(V). Define a symmetric bilinear form β(x, y) = Tr(φ(x)φ(y)). Let x_{1}, …, x_{n} be a basis for 𝔤 and let y_{1}, …, y_{n} be the dual basis with respect to the form β. The Casimir element is

c_{φ} = ∑ φ(x_{i})φ(y_{i}) ∈ End(V).

Proposition 2.6.3. c_{φ} commutes with φ(𝔤).

If φ is irreducible (in addition to being faithful) then by Schur’s lemma it follows that c_{φ} must be a scalar. Since

Tr(c_{φ}) = ∑ Tr(φ(x_{i})φ(y_{i})) = dim 𝔤

we see that c_{φ} = dim 𝔤/ dim V.

2.7. REPRESENTATIONS OF 𝔖𝔏(2) 23

2.6.3 Weyl’s theorem

Theorem 2.6.4 (Weyl). Any finite dimensional representation of a semisimple lie algebra is completely reducible.

Theorem 2.6.5. Let 𝔤 be a semisimple lie algebra, and φ : 𝔤 → 𝔤𝔩(V) a finite dimensional representation. If x = x_{s} + x_{n} is the abstract Jordan decomposition of x then φ(x_{s}) + φ(x_{n}) is the usual Jordan decomposition of φ(x) ∈ 𝔤𝔩(V).

2.7 Representations of 𝔰𝔩(2)

The lie algebra 𝔰𝔩(2) has a basis h = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ), x = ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ), y = ( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} ), so that [h, x] = 2x, [h, y] = −2y, and [x, y] = h.

Suppose V is an 𝔰𝔩(2)-module. Since h is diagonal the above theorem implies that h acts diagonally on V. Thus V is a direct sum of eigenspaces:

V = ⊕_{λ∈Λ}V_{λ}

where V_{λ} is the eigenspace (also called weight space) of V with eigenvalue (also called weight) λ ∈ k. We can completely describe the irreducible representations of 𝔰𝔩(2).

Theorem 2.7.1. Let V be an irreducible 𝔤 = 𝔰𝔩(2) module. Then

(a) If m = dim V − 1 then

V = V_{m} ⊕ V_{m−2} ⊕ ⋯ ⊕ V_{−(m−2)} ⊕ V_{−m}

and dim V_{m−2i} = 1 for each i.

(b) The action of 𝔤 on V is given explicitely by the following formulas. Choose a nonzero vector v_{0} ∈ V_{m}; set v_{−1} = 0, v_{i} = (1/i!)y^{i}.v_{0} (i ≥ 0).

(1) h.v_{i} = (m − 2i)v_{i},

(2) y.v_{i} = (i + 1)v_{i+1},

(3) x.v_{i} = (m − i + 1)v_{i−1} (i ≥ 0).

2.8 The root system arising from a toral decomposition

A toral subalgebra 𝔥 ⊂ 𝔤 is a subalgebra all of whose elements are semisimple.

Lemma 2.8.1. Any toral subalgebra 𝔥 ⊂ 𝔤 is abelian.

Proof. Suppose x ∈ 𝔥, then since ad_{𝔥}(x) is diagonalizable it suffices to show that the only eigenvalue of ad_{𝔥}(x) is 0. Suppose y ∈ 𝔱 is a (nonzero) eigenvector of ad_{𝔥}(x) so that ad_{𝔥}(x)(y) = λy for some λ ∈ k. Then ad(y)(x) = [y, x] = −[x, y] = − ad(x)(y) = −λy is an eigenvector of ad(y) with eigenvalue 0. Since ad(y) is diagonalizable it is possible to write x as a sum of eigenvectors of ad(y), x = ∑_{α} a_{α}v_{α}, with v_{α} an eigenvector of ad(y) with eigenvalue α. Then ad(y)(x) = ∑_{α} αa_{α}v_{α} = −λy. Since 𝔱 is a direct sum of the eigenspaces of ad(y) and y lies in the zero eigenspace it follows that, for each α, if a_{α} ≠ 0 then α = 0. But this exactly means that −λy = 0 and hence λ = 0, as claimed.

Let 𝔥 be a maximal toral subalgebra of 𝔤. Then 𝔥 acts on 𝔤 via the adjoint representation. Since 𝔥 is a commuting family of semisimple endomorphisms its action on 𝔤 is simultaneously diagonalizable so

𝔤 = ⨁_{α∈𝔥^{∗}} 𝔤_{α}

where

𝔤_{α} = {x ∈ 𝔤 : h.x = α(h)x for all h ∈ 𝔥 }.

By the lemma 𝔥 ⊂ 𝔤_{0}. In fact there is equality.

Proposition 2.8.2. 𝔤_{0} = 𝔥

Another basic fact is that the 𝔤_{α} satisfy certain strong orthogonality properties with respect to the Killing form.

Proposition 2.8.3. Let α, β ∈ 𝔥^{∗} and suppose α + β ≠ 0. Then κ(𝔤_{α}, 𝔤_{β}) = 0.

Proof. Choose h ∈ 𝔥 such that (α + β)(h) ≠ 0. Then if x ∈ 𝔤_{α} and y ∈ 𝔤_{β}, associativity of the Killing form gives κ([h, x], y) = −κ([x, h], y) = −κ(x, [h, y]), so α(h)κ(x, y) = −β(h)κ(x, y), or (α + β)(h)κ(x, y) = 0. This forces κ(x, y) = 0.

Proposition 2.8.4. The 𝔤_{α} (α ≠ 0) all have dimensional ≤ 1. If 𝔤_{α} ≠ 0 then 𝔤_{−α} ≠ 0. Furthermore, 𝔤_{cα} = 0 for all constants c ≠ 0, −1, 1.

Let Φ ⊂ 𝔥^{∗} be the set of nonzero (generalized) eigenvalues for the action of 𝔥 on 𝔤. The Killing form restricted to 𝔥 is nondegenerate and gives rise to a nondegenerate bilinear form β on 𝔥^{∗}. The roots Φ must span 𝔥^{∗}. Upon choosing a basis α_{1}, …, α_{n} ∈ Φ for 𝔥^{∗} it can be seen that each element of Φ is a ℚ-rational linear combination of the α_{i}. Furthermore β is positive definite as a bilinear form on the ℚ-rational span of the α_{i}. Thus, upon tensoring with ℝ, we obtaining a collection of vectors Φ inside a real Euclidean space E. The pair (Φ, E) forms a “root system” because it satisfies the properties enumerated in the following theorem.

Theorem 2.8.5. Let 𝔤, 𝔥, Φ, and E be as above. Then

• Φ spans E and 0 ∉ Φ.

• If α ∈ Φ then −α ∈ Φ but no other scalar multiple of α is in Φ.

• If α, β ∈ Φ, then

β − \frac{2(β, α)}{(α, α)}α ∈ Φ.

• If α, β ∈ Φ, then

\frac{2(β, α)}{(α, α)} ∈ ℤ.

Exercise 2.8.6 (II.8.4). Prove that each maximal toral subalgebra of 𝔰𝔩(2) is one dimensional.

Proof. 𝔰𝔩(2) is semisimple so nonzero maximal toral subalgebras must exist. Since toral subalgebras are abelian it suffices to show that 𝔰𝔩(2) has no 2 dimensional abelian subalgebras. Suppose a, b ∈ 𝔰𝔩(2) span a 2 dimensional abelian subalgebra. Let c ∈ 𝔰𝔩(2) such that a, b, c together span 𝔰𝔩(2). Then 𝔰𝔩(2) = [𝔰𝔩(2), 𝔰𝔩(2)] is spanned by [a, b] = 0, [a, c], and [b, c], which is absurd.

Exercise 2.8.7 (II.8.10). Prove that there are no four, five, or seven dimensional semisimple lie algebras.

Proof. First suppose a semisimple lie algebra 𝔤 has dimension 4. Then 𝔤 = 𝔥, or 𝔤 = 𝔥⊕𝔤_{α}⊕𝔤_{−α}, or 𝔤 = 𝔤_{α_{0}} ⊕𝔤_{−α_{0}} ⊕𝔤_{α_{1}} ⊕𝔤_{−α_{1}}. The first case is impossible because 𝔤 is not abelian. The second case is impossible because then 𝔥^{∗} has dimension 2 so Φ = {α, −α} can not span it. The third case is impossible because nonzero toral subalgebras always exist.

Next suppose 𝔤 has dimension 5. Let h = dim 𝔥 and let r = #Φ. Then r + h = 5 because

𝔤 = 𝔥 ⊕ ⨁_{α∈Φ} 𝔤_{α}.

Furthermore r is even and h ≤ r/2 since Φ contains a basis for 𝔥^{∗} which has dimension h. If h = 1 then r = 4 so Φ consists of 4 vectors, all lieing on a line through the origin. This contradicts the fact that if α ∈ Φ then −α is the only other scalar multiple of α which lies in Φ. If h = 2 then r = 3 which is impossible because r is even. If h ≥ 3 then r ≤ 2 which is impossible because then h is not ≤ r/2.

A similar argument as above applies to the case dim 𝔤 = 7.

2.9 Root systems

Let 𝔼 denote a finite dimensional vector space equipped with a positive definite symmetric bilinear form.

Definition 2.9.1. A root system is a a subset Φ ⊂ 𝔼 such that the following four axioms are satisfied:

(R1) Φ is finite, spanes 𝔼, and 0 ∉ Φ.

(R2) The only multiples of α ∈ Φ are ±α.

(R3) If α ∈ Φ the reflection σ_{α} leaves Φ invariant.

(R4) If α, β ∈ Φ, then

⟨β, α⟩ = \frac{2(β, α)}{(α, α)} ∈ ℤ.

Proposition 2.9.2. Let α ∈ 𝔼 be nonzero, then σ_{α}, the reflection through the hyperplane orthogonal to α, is given explicitely by

σ_{α}(β) = β − ⟨β, α⟩α.

Proof. This maps fixes the hyperplane orthogonal to α and sends α to −α. Thus it agrees with σ_{α} on a basis and hence must agree with σ_{α}.

Definition 2.9.3. Let Φ ⊂ 𝔼 be a root system. The Weyl group W of Φ is the subgroup of GL(𝔼) generated by the reflections σ_{α} with α ∈ Φ.

Since Φ spans 𝔼, W can be viewed as a subgroup of the group of permutations of Φ. In particular, W is finite.

Proposition 2.9.4. Let Φ be a root system in 𝔼, with Weyl group W. If σ ∈ GL(𝔼) leaves Φ invariant, then

σσ_{α}σ^{−1} = σ_{σ(α)}

for all α ∈ Φ.

Proposition 2.9.5. Let Φ be a root system, and suppose α, β ∈ Φ are such that (α, β) > 0. Then α − β ∈ Φ.

Proof. The cosine of the angle θ between α and β is given by the formula ||α||||β|| cos θ = (α, β). Therefore,

⟨β, α⟩ = 2\frac{||β||}{||α||} cos θ

hence ⟨β, α⟩⟨α, β⟩ = 4 cos^{2} θ. Because (α, β) > 0 both ⟨β, α⟩ and ⟨α, β⟩ are > 0 and because they are both integers and 0 < cos^{2} θ ≤ 1, their product must be either 1, 2, or 3. Thus either ⟨β, α⟩ or ⟨α, β⟩ must equal 1. If ⟨α, β⟩ = 1 then σ_{β}(α) = α−β ∈ Φ. If ⟨β, α⟩ = 1 then β −α ∈ Φ so α − β ∈ Φ.

Exercise 2.9.6 (III.9.1). Let 𝔼′ be a subspace of 𝔼. If a reflection σ_{α} leaves 𝔼′ invariant, prove that either α ∈ 𝔼′ or else 𝔼′ ⊂ α^{⊥}.

Proof. Suppose 𝔼′ is not a subset of α^{⊥}. We must show that α ∈ 𝔼′. By assumption there exists x ∈ 𝔼′ such that (x, α) ≠ 0. Since σ_{α} leaves 𝔼′ invariant we have σ_{α}(x) = x− \frac{(x,α)}{(α,α)}α ∈ 𝔼′.

Since 𝔼′ is a subspace x − σ_{α}(x) = \frac{(x,α)}{(α,α)}α ∈ 𝔼′. Since (x, α) ≠ 0 it follows that α ∈ 𝔼′, as claimed.

Algebraic Number Theory

3.1 Trace map

3.2 The Discriminant and Ramification

********* Begin Verified *********

Definition 3.2.1. Let K be a number field of degree n and choose an integral (i.e., ℤ-module) basis {ω_{1}, …, ω_{n}} for O_{K}. Such a basis exists because O_{K} is a free ℤ-module whose rank must be n. Let \{σ_{1}, …, σ_{n}\} be the distinct embeddings of K into \overline{ℚ}. Then the discriminant of K is

D_{K} = det(σ_{i}ω_{j})^{2} ∈ ℤ

The discriminant is well-defined because making another choice of integral basis multiplies D_{K} by the square of an element of ℤ^{×}, i.e., by 1.

Definition 3.2.2. An extension L/K of local fields is ramified if the extension O_{L}/O_{K} of integer rings is ramified. The extension ℂ/ℝ is ramified.

Theorem 3.2.3. Let L/K be an unramified extension of local fields. Then

N_{L/K}(O_{L}^{×}) = O_{K}^{×}.

Proof. Page 50 of [6]. [There is probably a slick Galois cohomology proof which I should work out.]

Consider for example the extension ℚ_{2}(i)/ℚ_{2}. Then −1 ∈ ℤ_{2}^{×} can not be the norm of an element of ℤ_{2}[i]^{×} because a^{2} + b^{2} = −1 has no solution mod 4.

Definition 3.2.4. Let L/K be an extension of number fields. Then a place v of K is said to ramify in L/K if the extension L_{w}/K_{v} is ramified for some place w of L lying over v.

Theorem 3.2.5. The primes p which ramify in K are exactly the primes which divide D_{K}.

Definition 3.2.6. Two extensions K and L of a field F are linearly disjoint if K ⊗_{F} L is a field.

If K and L are number fields with fixed embeddings K, L ⊂ \overline{ℚ} then they are linearly disjoint iff

[K.L : ℚ] = [K : ℚ][L : ℚ].

To see this consider the natural surjection K ⊗_{ℚ} L → K.L and count dimensions.

Remark 3.2.7. If K ∩ L = ℚ then K and L need not be linearly disjoint. For example, let K = ℚ(2^{\frac{1}{3}} ) and L = ℚ(ζ_{3}, 2^{\frac{1}{3}} ). Then K ∩ L = ℚ but [K.L : ℚ] = 6 so K and L are not linearly disjoint.

If K and L are both Galois then K ∩L = ℚ iff K and L are linearly disjoint. This is because the Galois group of K.L is the product Gal(K/ℚ) × Gal(L/ℚ).

Definition 3.2.8. The Galois closure of a number field K is the minimal Galois subfield of \overline{ℚ} which contains K. This is the same as the compositum of all Galois conjugates of K.

Proposition 3.2.9. The primes which ramify in K are the same as the primes which ramify in the Galois closure L of K.

Proof. The primes which ramify in K are (clearly) a subset of the primes which ramify in L. Suppose the prime p ∈ ℤ is unramified in K. We must show that p is unramified in L. Let H ⊂ G = Gal(L/ℚ) be the subgroup corresponding to K by Galois theory. Let w be any place of L lying above p and let v be its restriction to K. Since e(w/p) = e(w/v)e(v/p) and p does not ramify in K (so e(v/p) = 1) we see that e(w/p) = e(w/v). The inertia group I_{w/p} of w has order e(w/p) = e(w/v). Since H is the Galois group of L/K we see that I_{w/v} = I_{w/p} ∩ H. Thus

[I_{w/p} ∩ H] = [I_{w/v}] = [I_{w/p}]

so I_{w/p} ⊂ H. Since p is unramified in any Galois conjugate of K the same argument shows that any conjugate of H also contains I_{w/p}. But L is the Galois closure of K so

I_{w/p} ⊂ ∩_{σ∈G}σHσ^{−1} = {1}

which shows that p does not ramify in L.

Proposition 3.2.10. If number fields K and L have relatively prime discriminants then they are linearly disjoint.

Proof. If K and L have relatively prime discriminants then so do their Galois closures, so we may assume that K and L are Galois. Since there are no nontrivial unramified extensions of ℚ and K ∩ L is an unramified extension of ℚ, we must have K ∩ L = ℚ. Because K and L are Galois they must be linearly disjoint.

Theorem 3.2.11. If K, L ⊂ \overline{ℚ} are number fields whose discriminants are relatively prime then

O_{KL} = O_{K}O_{L}

and

D_{KL} = D_{K}^{[L:ℚ]}D_{L}^{[K:ℚ]}.

Proof. [6][III,3]

Example 3.2.12. Let K = ℚ(\sqrt{−1}) and F = ℚ(\sqrt{3}). Then D_{K} = −4, D_{F} = 12, and D_{KF} = 144 (computed with pari). Thus the condition that (D_{K}, D_{F}) = 1 is necessary in the previous theorem.

Example 3.2.13. Suppose K, L ⊂ ℚ are number fields and that K′ ∩ L = ℚ for all conjugates K′ of K. Let F be the Galois closure of K. Must F ∩ L = ℚ? No! For example, consider K = ℚ(2^{1/3}) and L = ℚ(ζ_{3}). L is quadratic and all conjugates of K are cubic so K′ ∩ L = ℚ. But the Galois closure of K contains L.

28 CHAPTER 3. ALGEBRAIC NUMBER THEORY

3.3 The Minkowski Bound

********* Begin Verified *********

Theorem 3.3.1. Let K/ℚ be a number field having degree N, discriminant D_{K}, and r_{2} complex places. Every ideal class in Pic(O_{K}) contains an ideal of norm less than or equal to

\frac{N!}{N^{N}} (\frac{4}{π} )^{r_{2}} · \sqrt{|D_{K}|}.

Proof. See [6] [V,4]. Example 3.3.2. Let K = ℚ(\sqrt{3}). Then N = 2, D_{K} = 12, and r_{2} = 0 so every ideal class in Pic(O_{K}) contains an ideal of norm less than or equal to

\frac{2!}{2^{2}} (\frac{4}{π} )^{0} · \sqrt{12} = \sqrt{3} ≈ 1.73

Thus Pic(O_{K}) is trivial.

********* End Verified *********

3.4 Cyclotomic Fields

********* Begin Unverified *********

Theorem 3.4.1. Let K = ℚ(ζ_{p^{r}}) where ζ_{p^{r}} is a primitive p^{r}-th root of unity. Then • Gal(K/ℚ) ≅ (ℤ/p^{r}ℤ)^{×},

• O_{K} = ℤ[ζ_{p^{r}}],

• p is totally ramified in K, and

• D_{K} = ±p^{p^{r−1}(pr−r−1)} (where the − sign holds when p^{r} = 4 or p ≡ 3 (mod 4), and the +

sign holds otherwise.) Proposition 3.4.2. The group (ℤ/p^{r}ℤ)^{×} is cyclic when p > 2 and is generated by −1 and 5 when p = 2.

********* End Unverified *********

3.5 Hilbert’s 90

********* Begin Unverified *********

Let L/K be a finite Galois extension. Proposition 3.5.1. For every integer q, Ĥ^{q}(L/K, L) = 0. Proposition 3.5.2. H^{1}(L/K, L^{×}) = 0.

********* End Unverified *********

3.6 Kummer Theory

********* Begin Unverified *********

Let K be a number field. Then class field theory allows us to see the abelian extensions of K by looking at the ideles J_{K}. Suppose we are interested only in seeing the abelian extensions of exponent dividing a fixed positive integer n. When K contains the n-th roots of unity, extensions obtained by adjoining an n-th root of an element of K are called Kummer extensions. Such extensions are relatively easy to understand directly in terms of the arithmetic of K. Thus we can see the exponent n abelian extensions because they are built up out of Kummer extensions. Understanding Kummer extensions is a key step in proving the existence theorem of class field theory.

Assume that μ_{n} ⊂ K. Taking Galois cohomology of the exact sequence

0 → μ_{n} → K̄^{×} ⟶^{x↦x^{n}} K̄^{×} → 0

gives rise to an exact sequence

0 → K^{×}/(K^{×})^{n} ⟶^{δ} H^{1}(K, μ_{n}) → H^{1}(K, K̄^{×}).

By Hilbert’s Theorem 90, H^{1}(K, K̄^{×}) = 0 so there is an isomorphism

K^{×}/(K^{×})^{n} ⟶^{≅} H^{1}(K, μ_{n}) = Hom(Gal(K̄/K), μ_{n}) = Hom(Gal(K_{n}/K), μ_{n})

where K_{n} is the compositum of all abelian extensions of K whose exponent divides n. Therefore the dual of the galois group of interest to us is K^{×}/(K^{×})^{n}. Furthermore there is a correspondence

\{ \begin{matrix}abelian extensions L/K \\ of exponent dividing n \end{matrix} \} ⟶^{≅} \{ finite subgroups G ⊂ K^{×}/(K^{×})^{n}\}

in which

L ↦ H^{1}(L/K, μ_{n}) ↪ H^{1}(K, μ_{n}) ≅ K^{×}/(K^{×})^{n}

and a subgroup G of K^{×}/(K^{×})^{n} corresponds to the field K( \sqrt{G}^{\frac{1}{n}}).

[Question: In Serre’s Local Fields it is stated that there is a canonical isomorphism

Gal(K_{n}/K) ≅ μ_{n} ⊗ Hom(K^{×}/(K^{×})^{n}, μ_{n})

but I don’t understand why this is.]

********* End Unverified *********

3.7 Hensel’s Lemma

********* Begin Unverified *********

Theorem 3.7.1. Let K be a field complete with respect to a non-archimedian valuation | · | and let f(x) ∈ O_{K}[x]. Let α_{0} ∈ O_{K} such that |f(α_{0})| < |f′(α_{0})|^{2}. Then there exists α ∈ O_{K} such that f(α) = 0.

Proof. Let

For example if f(α_{0}) ≡ 0 (mod 𝔪) but f′(α_{0}) ≢ 0 (mod 𝔪) then f(x) has a root in O_{K}.

3.8 Ideles and adeles

3.8.1 Restricted products

Definition 3.8.1. Let {X_{λ}} be a collection of topological spaces and, for all but finitely many λ, let U_{λ} be an open subset of X_{λ}. Then the restricted direct product of the {X_{λ}} with respect to the U_{λ}, denoted ∏′_{λ} X_{λ} is defined as follows. As a set it is the collection of sequences (x_{λ}) where x_{λ} ∈ X_{λ} and x_{λ} ∈ U_{λ} for all but finitely many λ. A basis for open sets is formed by taking all sets of the form ∏_{λ} V_{λ} where V_{λ} ⊂ X_{λ} is an open set and V_{λ} = U_{λ} for all but finitely many λ.

Lemma 3.8.2. Let Λ be a finite set of indices. For each λ ∈ Λ let D_{λ} be a closed subset of X_{λ}. Then the set

∏_{λ∈Λ} D_{λ} × ∏_{λ∉Λ}^{′} U_{λ}

is a closed subset of ∏.

Proof. a

Definition 3.8.3. A topological space X is locally compact if for every point x ∈ X there is a compact subset D and an open set U such that x ∈ U ⊂ D. In other words, every point has a compact neighborhood.

Proposition 3.8.4. Suppose that the X_{λ} are locally compact and that the U_{λ} are compact. Then ∏ = ∏′_{λ} X_{λ} is locally compact.

Proof. Let (x_{λ}) ∈ ∏. Let

Λ = {λ : x_{λ} ∉ U_{λ}}.

Since each X_{λ} is locally compact, for each λ ∈ Λ there is a closed subset D_{λ} of X_{λ} and an opened subset U′_{λ} ⊂ X_{λ} such that x_{λ} ∈ U′_{λ} ⊂ D_{λ}. Then the product

∏_{λ∈Λ} D_{λ} × ∏_{λ∉Λ}^{′} U_{λ}

is a closed subset of ∏ which contains (x_{λ}) and which contains an open set containing (x_{λ}).

Galois Theory

4.1 Separability

Definition 4.1.1. An algebraic extension L/K of fields is separable if the minimal polynomial of every element x ∈ L has distinct roots in some fixed algebraic closure K̄ of K. The separable degree of a finite algebraic extension L/K, denoted [L/K]_{sep} is the number of distinct Kembeddings of L into K̄.

Theorem 4.1.2 (Primitive Element Theorem). Let L/K be a finite separable extension of fields. Then L = K(α) for some α ∈ L.

Example 4.1.3. The primitive element theorem may fail when L/K is not separable. For example, let k be a field of characteristic p and let K = k(s, t) where s and t are independent transcendentals. Let L = K(s^{\frac{1}{p}} , t^{\frac{1}{p}}). Then L/K is purely inseparable of degree p^{2} since L^{p} = K (note that p-th powering is an homomorphism and the p-th powers of basis elements of L lie in K). Thus if L = K(α) for some α ∈ L then α^{p} ∈ K so p^{2} = [L/K] ≤ p, which is absurd. Thus there can be no primitive element for L/K. Replacing s and t by n transcendentals s_{1}, …, s_{n} shows that an arbitrarily large number of generators may be necessary. Another interesting remark is that there must be infinitely many distinct fields F with K ⊂ F ⊂ L. In fact all fields of the form F = K(s^{\frac{1}{p}} +ct^{\frac{1}{p}}) must be distinct because if two were the same then it would follow that F = L because both s^{\frac{1}{p}} and t^{\frac{1}{p}} would be forced to lie in F. It is also interesting to note that if k is replaced by a field of characteristic ≠ p then the extension L/K defined above must be separable and hence there must be a primitive element. In fact, s^{\frac{1}{p}} +t^{\frac{1}{p}} is a primitive element because it has p^{2} distinct conjugates in characteristic ≠ p (because the polynomial x^{p} − 1 has distinct roots in characteristic ≠ p.)

On the other hand, there exist extensions generated by a single element which are not separable. For example, let k be a field of characteristic p and let s be transcendental over k. Then the extension k(s^{\frac{1}{p}})/k(s) is generated by a single element but is not separable.

Number theory qual review

5.1 Discriminant

5.1.1 Introduction

Let L/K be a degree n separable extension of fields. Let σ_{1}, …, σ_{n} be the distinct Kembeddings of L into K̄. Let ω_{1}, …, ω_{n} be n elements of O_{L}. Then the discriminant of ω_{1}, …, ω_{n} is

D_{L/K}(ω_{1}, …, ω_{n}) = (det(σ_{i}ω_{j}))^{2}.

Because applying some σ_{i} to the matrix (σ_{i}ω_{j}) permutes the rows we see that the sign of the determinent is the only thing which changes, hence D_{L/K} is fixed by all σ_{i}, so D_{L/K} ∈ O_{K}.

Suppose K is a number field. Then O_{K} is a torsion free ℤ-module hence free so there exists a basis ω_{1}, …, ω_{n} for O_{K} over ℤ. The discriminant of K is the discriminant of this basis. Choosing a different integral basis changes the discrimant by the square of a unit in ℤ, i.e., it does not change the discriminant. Thus the discriminant is a well defined element of ℤ.

Suppose L/K is an extension of number fields. Then O_{L} may or may not be a free O_{K}- module. Suppose O_{L} is a free O_{K}-module with basis ω_{1}, …, ω_{n}. Then D_{L/K}(ω_{1}, …, ω_{n}) is an element of O_{K}, well defined up to the square of a unit in O_{K}.

If O_{L} is not free as an O_{K}-module then there is still a way to define the discriminant D_{L/K}. Instead of being an element of O_{K}, the discriminant is an integral ideal in O_{K}. It is divisible by exactly those primes which ramify in L/K. To define this more general notion of discriminant we need the notion of the module index and trace form.

5.1.2 The module index

Definition 5.1.1. Let O be a Dedekind domain and let V be a finite dimensional vector space over the quotient field of O. Let M and N be O modules which are lattices in V. Then the module index of N in M is the fractional O-ideal [M : N] defined as follows. For each prime 𝔭 of O, O_{𝔭} is a discrete valuation ring hence M_{𝔭} and N_{𝔭} are free O_{𝔭}-modules of the same rank, hence isomorphic. Thus there exists a linear transformation ℓ : V → V such that ℓ(M_{𝔭}) = N_{𝔭}. Then the 𝔭 component of [M : N] is det(ℓ)O_{𝔭}.

Note that the isomorphism ℓ is an isomorphism of O_{𝔭}-modules but, upon choosing a basis for N_{𝔭}, the matrix ℓ need not have coefficients in O_{𝔭}. For example, let M = pℤ_{p} and N = ℤ_{p} and let ℓ be the division-by-p isomorphism. Then, with respect to the basis 1 for N, the matrix of ℓ is (\frac{1}{p}) (view M and N as sitting in ℚ_{p}.)

The module index is well defined because ℓ is defined up to an automorphism of M_{𝔭} and hence a different choice of ℓ changes det(ℓ) by a unit in O_{𝔭}. Also, it can be shown that there are only finitely many primes 𝔭 such that det(ℓ) is not a unit. In the situation when M and N are already free as O-modules then ℓ can be chosen globally and then the module index is just det(ℓ)O.

Example 5.1.2. Let O be a Dedekind domain with quotient field K and let 𝔞_{1}, 𝔞_{2} be nonzero fractional O-ideals. Thus 𝔞_{1} and 𝔞_{2} are lattices in K. The module index [𝔞_{1} : 𝔞_{2}] is the fractional ideal 𝔞_{1}^{−1}𝔞_{2}. To see this localize at a prime 𝔭 of O. Then (𝔞_{1})_{𝔭} = 𝔭^{a_{1}} and (𝔞_{2})_{𝔭} = 𝔭^{a_{2}} so we may take ℓ = π^{a_{2}−a_{1}} where π is a uniformizing parameter for 𝔭. Taking O to be the ring of integers of a number field with class number > 1 this gives examples where the module index cannot be computed globally.

Example 5.1.3. Let O = ℤ[α] where α^{2} + α + 1 = 0. Let M = ℤ[α] = ℤ[\frac{−1+\sqrt{−3}}{2}] and let N = ℤ[2α + 1] = ℤ[\sqrt{−3}]. We view M and N as ℤ-modules and compute the module index [M : N]. Both M and N are free ℤ-modules so there is an isomorphism ℓ : M → N. With respect to the basis 1, α, it is represented by the matrix ( \begin{matrix}1 & 1 \\ 0 & 2 \end{matrix} ). Thus [M : N] = det(ℓ)ℤ = 2ℤ. Note that M/N ≅ ℤ/2ℤ.

5.1.3 Trace form

Definition 5.1.4. Let K be a field and let A be a finite dimensional central K-algebra. Left multiplication by an element x ∈ A induces a K-linear map ℓ_{x} : A → A. The trace of x, denoted tr(x), is the trace of of ℓ_{x}. Since ℓ_{x} is an endomorphism of a K-vector space, tr(x) ∈ K.

We will consider the trace in the case when L = A is a finite extension field of K. The degree to which the trace is nondegenerate is reflected in the degree to which the extension L/K is separable. The trace also gives rise to a symmetric bilinear trace form ⟨α, β⟩_{L/K} = tr(αβ) which, among other things, provides an alternative way to define and compute discriminants. The trace form is like the Killing form in the theory of lie algebras. Just as the trace form is nondegenerate iff the field extension L/K is separable, the Killing form on a lie algebra 𝔤 is nondegenerate (at least in characteristic 0) iff 𝔤 is semisimple. Also, the map x → ℓ_{x} is exactly analogous to the adjoint representation.

Lemma 5.1.5. Suppose A/K is a central K-algebra of dimension n. Let x ∈ A and let χ(t) ∈ K[t] be the characteristic polynomial of ℓ_{x}, so χ(t) = det(t − ℓ_{x}). Then tr(x) is minus the coefficient of t^{n−1}.

Proof. It is a general fact that the trace of an endomorphism τ of a vector space V (of dimension n) is minus the coefficient of t^{n−1} in the characteristic polynomial of τ. This is can be seen at once by considering the Jordan canonical form of τ.

Lemma 5.1.6. Suppose L/F/K is a tower of fields and L is separable over K. Then

1. L/F and F/K are both separable.

2. Let {τ_{1}, …, τ_{n}} be the distinct F -embeddings of L and let {σ_{1}, …, σ_{m}} be the distinct K-embeddings of F. Then the distinct K-embeddings of L are

{τ_{i}σ_{j} : i = 1, …, n, j = 1, …, m}.

(Extend σ_{i} to L by letting it act as the identity on a basis for L/F.)

Proof. The collection of K-embeddings of L, upon restricting to F, give the K-embeddings σ_{1}, …, σ_{m} of F. Each of these can be extended to a K-embedding of L in at most [L : F] ways. We know that there are

[L : K] = [L : F][F : K]

K-embeddings of L. Thus [F : K]_{sep} = m = [F : K] and each K-embedding of F must extend to a K-embedding of L in [L : F] ways. In particular, there are [L : F] K-embeddings of L extending the identity embedding of F, i.e., [L : F]_{sep} = [L : F]. Thus part (1) is proved and part (2) follows at once.

Theorem 5.1.7. Suppose L/F/K is a tower of finite extensions of fields. Then tr_{L/K} = tr_{F/K} ∘ tr_{L/F}.

Proof.

Proposition 5.1.8. Suppose L/K is a finite separable extension of fields and let σ_{1}, …, σ_{n} be the distinct K embeddings of L into a fix algebraic closure K̄ of K. If α ∈ L then

tr(α) = σ_{1}(α) + ⋯ + σ_{n}(α).

Proof. Let F = K(α). Let f(x) be the minimal polynomial of α over K and let χ(x) be the characteristic polynomial of α acting on the vector space F. By the Cayley-Hamilton theorem χ(α) = 0 so f must divide χ. But deg(f) = [F : K] = deg(χ) so f = χ. The trace of α is minus the coefficient of x^{[F:K]−1} in χ(x). But f(x) = ∏(x − σ(α)) where the product is over the distinct K-embeddings σ of F into a fixed K̄, so the coefficient of x^{[F:K]−1} is minus the sum of the conjugates of σ(α). Thus the proposition is true for tr_{F/K}(α).

Finally,

tr_{L/K}(α) = tr_{F/K}(tr_{L/F} (α)) = tr_{F/K}([L : F]α) = [L : F] tr_{F/K}(α) = ∑ σ_{i}(α)

because of how the K-embeddings of L are made up of the F -embeddings of L and the Kembeddings of F, as described in the above lemma.

Theorem 5.1.9. Let L/K be a finite separable extension of fields and suppose ω_{1}, …, ω_{n} ∈ L. Then

D_{L/K}(ω_{1}, …, ω_{n}) = det(⟨ω_{i}, ω_{j}⟩)

where ⟨ω_{i}, ω_{j}⟩ = tr_{L/K}(ω_{i}ω_{j}).

Proof.

Theorem 5.1.10. Suppose L/K is a finite separable extension of fields. Then the trace form

⟨ , ⟩_{L/K} : L × L → K

is nondegenerate.

Proof.

Theorem 5.1.11. Suppose L/K is a finite purely inseparable extension of fields. Then tr : L → K is the zero map.

Proof.

Homological Algebra

6.1 Inverse and Direct Limits

Theorem 6.1.1. Let R be a ring and let M, N, and {M_{i}} be modules over R, such that the collection {M_{i}} form an inverse or directed system, as appropriate. Then

Hom(N, lim_{←} M_{i}) ≅ lim_{←} Hom(N, M_{i})
Hom(lim_{→} M_{i}, N) ≅ lim_{→} Hom(M_{i}, N)

N ⊗ lim_{→} M_{i} ≅ lim_{→}(N ⊗ M_{i})

Proof. Direct limits commute with left adjoints and inverse limits commute with right adjoints. The functor Hom(N, −) is right adjoint to the left adjoint functor N ⊗ − because there is a natural isomorphism

Hom(M, Hom(N, T)) ≅ Hom(N ⊗ M, T).

Galois and Group Cohomology

7.1 Order and Index

********* Begin Unverified *********

Let X be a curve over a perfect field k. Let Div(X) be the free abelian group generated by the closed points of X. A prime divisor is the sum consisting of a single closed point. Let K = k(X) be the function field of X over k (i.e., κ(ξ) where ξ is the generic point of X). There is a map μ_{X} : K^{×} → Div(X) which associates to a function f ∈ K^{×} its divisor

μ_{X}(f) = ∑_{P ∈#X} v_{P} (f)P ∈ Div(X)

where #X is the underlying space of the scheme X and v_{P} is the valuation on the local ring X_{P} of X at P. Divisors in the image of μ_{X}(f) are called principal divisors. [The valuation at the generic point is trivial.] Let Pic(X) be defined by the exact sequence

0 → K^{×} → Div(X) → Pic(X) → 0.

Define a homomorphism deg : Div(X) → ℤ by

deg(∑ n_{i}P_{i}) = ∑ n_{i}[κ(P_{i}) : k]

where the P_{i} are prime divisors.

Theorem 7.1.1. The degree of a principal divisor is 0.

The theorem implies that the degree map extends in a well-defined way to Pic(X). Denote by Div^{0}(X) (resp., Pic^{0}(X)) the kernel of the degree map. Let k̄ be an algebraic closure of k and let X̄ = X ×_{k} k̄.

Proposition 7.1.2. The natural map

Pic(X) ↪ H^{0}(k, Pic(X̄))

is injective.

Proof. Suppose D represents a class in Pic(X) which becomes principal in Pic(X̄). [I don’t know how to prove this.]

The index of X is the positive integer d such that

Div(X) ⟶^{deg} ℤ → ℤ/dℤ → 0

is exact, and the order of X is the positive integer n such that

H^{0}(k, Pic(X̄)) ⟶^{deg} ℤ → ℤ/nℤ → 0

is exact. Note that since Div(X) surjects onto Pic(X) we could have also defined the index of X by the exact sequence Pic(X) → ℤ → ℤ/dℤ → 0.

Consider the exact sequence of Gal(k̄/k)-modules

0 → Pic^{0}(X̄) → Pic(X̄) → ℤ → 0

in which ℤ is given the trivial action. Taking cohomology gives

H^{0}(k, Pic(X̄)) ⟶^{deg} H^{0}(k, ℤ) ⟶^{δ_{X}}→ H^{1}(k, Pic^{0}(X̄)).

Let J be the Jacobian of X. Fixing a structure of principal homogeneous space for X over J gives a functorial (in k) isomorphism (more on this later)

H^{0}(k, Pic^{0}(X̄)) ≅ H^{0}(k, J(k̄)).

Combining this with the fact that H^{0}(k, ℤ) = ℤ allows us to rewrite the above exact sequence as

H^{0}(k, Pic(X̄)) ⟶^{deg} H^{0}(k, ℤ) ⟶^{δ_{X}}→ H^{1}(k, J(k̄)).

Then the element δ_{X}(1) ∈ H^{1}(k, J(k̄)) is a class in the cohomology of the abelian variety J naturally (after choice of principal homogeneous space structure) associated to the curve X. [What exactly is meant by the phs structure when X has genus greater than 1?] Of course δ_{X}(1) is the same class as was associated to X in Lang-Tate because δ_{X}(1) is constructed by choosing a degree 1 element of Pic(X), i.e., a point P ∈ X(k̄) and considering the cohomology class in H^{1}(k, Pic^{0}(X̄)) given by the corresponding cocycle (σ ↦ σp − p).

To give a map Pic^{0}(X̄) → J(k̄) it suffices to give a map which sends differences P − Q to poitns on J(k̄) since every element of Pic^{0}(X̄) is represented by a sum of such differences P − Q with P, Q ∈ X(k̄). Given a principal homogeneous space structure J × X → X let P − Q map to the unique element v ∈ J(k̄) such that (v, Q) ↦ P, (i.e., P − Q = v iff P = Q + v.)

Proposition 7.1.3. The order of δ_{X}(1) is the same as the order of X as defined above. The index of the cohomology class δ_{X}(1) is the same as the index of X.

Proof. If the order of δ_{X}(1) is n then the cokernel of the map

H^{0}(k, Pic(X̄)) ⟶^{deg} H^{0}(k, ℤ) = ℤ

is ℤ/nℤ hence the order of X is n. Next note that δ_{X}(1) restricts to 0 in an extension ℓ/k iff δ_{X_{ℓ}}(1) = 0. This is equivalent to X_{ℓ} defining the trivial cohomology class which is in turn equivalent to X(ℓ) being nonempty. Thus the two definitions of index agree.

Corollary 7.1.4. The order of X divides the index of X and they have the same prime factors.

********* End Unverified *********

Theorem 7.1.5 (Basic Exact Sequence). There is an exact sequence

0 → Pic(X) → H^{0}(k, Pic(X̄)) → Br(k) → Br(X) → H^{1}(k, Pic(X̄)) → H^{3}(k, k̄^{×}).

Proof. [Enhance the “elementary” proof in Lichentbaum.]

These tools allow me to see, for a particular curve X, the relationship between the period and the index. But they give me little insight into the collection of all curves with a given jacobian, and they don’t allow me to see how often the period equals the index.

7.2 H^{1}(GL(V), V)

Theorem 7.2.1. If V is a vector space over a field K of characteristic not equal to 2 then

H^{1}(GL(V), V) = 0.

Proof. If dim V > 1 we consider the exact sequence

0 → SL(V) → GL(V) ⟶^{det} K^{×} → 0.

It gives rise (via the Hochschild-Serre spectral sequence) to the exact sequence

0 → H^{1}(K^{×}, V^{SL(V)}) ⟶^{inf} H^{1}(GL(V), V) ⟶^{res} H^{1}(SL(V), V)^{GL(V)/ SL(V)}

Since SL(V) has no fixed points (since dim V > 1) we see that H^{1}(K^{×}, V^{SL(V)}) = 0 and hence res is injective. Let 2 ∈ GL(V) be the multiplication-by-2 linear automorphism. Then the image of 2 in GL(V)/ SL(V) fixes the image (under res) of H^{1}(GL(V), V). But if f : SL(V) → V is a cocycle then

2 ∗ f(σ) = 2f(2^{−1}σ2) = 2f(σ).

Thus if c is in the image of H^{1}(GL(V), V) then 2c = c and hence c = 0. Since res is injective we conclude that H^{1}(GL(V), V) = 0.

If dim V = 1 we use the exact sequence

0 → {±1} → K^{×} → (K^{×})^{2} → 0.

This gives rise to the exact sequence

0 → H^{1}((K^{×})^{2}, K^{{±1}}) ⟶^{inf} H^{1}(K^{×}, K) ⟶^{res} H^{1}({±1}, K)

Since −1 has no nonzero fixed points, K^{{±1}} = 0 so res is injective. Since {±1} has order 2, H^{1}({±1}, K) is killed by 2. But multiplication by 2 is an automorphism on K so it induces an automorphism on H^{1}({±1}, K). Thus H^{1}({±1}, K) = 0.

Question 7.2.2. It is also true that H^{1}(GL(2, 𝔽_{2}), 𝔽_{2}^{2}) = 0 as can be seen by considering the exact sequence

0 → ⟨( \begin{matrix}1 & 1 \\ 1 & 0 \end{matrix} )⟩ → GL(2, 𝔽_{2}) → ℤ/2ℤ → 0.

What is the complete answer in characteristic 2?

7.3 Galois cohomology of abelian varieties

When A is an algebraic group over a field k, we now write π_{0}(A) for the set of connected components (for the Zariski topology) of A over k_{s}; that is, π_{0}(A) = A_{k_{s}}/A_{k_{s}}^{0} regarded as a G-module.

Theorem 7.3.1. Let A/K be an abelian variety with Néron model A/R. Then

H^{1}(K_{un}/K, A(K_{un})) = H^{1}(K_{un}/K, π_{0}(A_{0}))

where A_{0} is the special fibre of A/R. In particular, if A has good reduction, then H^{1}(K_{un}/K, A(K_{un})) =

7.4 Descent theory

********* Begin Unverified *********

Let k be a field. Suppose that E is an elliptic curve which can be defined over k but view E as an elliptic over the separable closure k^{sep} of k. Define a group B as follows. As a set B is the collection of all pairs (τ, σ) where τ : E → E is an isomorphism of schemes (not necessarily over k^{sep}) and σ is a k-automorphism of k^{sep} such that the following diagram commutes.

E  ⟶^{τ}  E
↓  ↓
Spec k^{sep} ⟵^{\overline{σ}} Spec k^{sep}

Here \overline{σ} denotes the morphism of schemes induced by σ. Note that the map induced by (τ, σ) from E(k^{sep}) → E(k^{sep}) is a group homomorphism. This can be seen by a slight generalization in which we consider such morphisms (τ, σ) between pairs (E, ℓ) where E is an elliptic curve over a field ℓ. Thus a morphism between (E, ℓ) and (F, m) is a pair (τ, σ) where τ : E → F is an isomorphism of schemes and σ : ℓ → m is an isomorphism of fields and the following diagram commutes.

E ⟶^{τ} F
↓  ↓
Spec ℓ ⟵^{\overline{σ}} Spec m

The induced morphism on points can then be thought of relabeling a point on E and relabeling the defining equations of E then mapping the point over to F by an isomorphism of elliptic curves over Spec ℓ where F is given the Spec ℓ structure induced by σ.

The product (τ, σ) · (τ′, σ′) is (τ′τ, σ′σ).

E ⟶^{τ} F ⟶^{τ′} G
↓  ↓  ↓
ℓ ⟵^{\overline{σ}} m ⟵^{\overline{σ}′} n

Recall that we have fixed an elliptic curve F over k^{sep}, and defined B to be the set of pairs (τ, σ) with τ : F → F an isomorphism of schemes and τ : k^{sep} → k^{sep} a field isomorphism over k. Let G = Gal(k^{sep}/k). Then there is an exact sequence

0 → Aut_{k^{sep}}(E) → B ⟶^{(τ,σ)↦σ} G → 0.

The projection map to B → G is surjective exactly because we assumed that F actually arises by base extension from some elliptic curve over k. Thus there is E/k such that F = E ×_{k} k^{sep}. Using E we see that given any automorphism σ ∈ 𝔾 we can construct an element of B. Simply take the product of the map σ̄ : Spec k^{sep} → Spec k^{sep} with the identity map 1 : E → E to obtain a morphism of schemes

F ≅ E ×_{k} k^{sep} ⟶^{1×σ̄} E ×_{k} k^{sep} ≅ F.

For notational convenience let A = Aut_{k^{sep}}(F). There is often an isomorphism

{E/k : E ≅_{k^{sep}} F }/ ≅_{k}⟶ { continuous group-theoretic sections G → B }/A.

Class Field Theory

8.1 Main theorems of class field theory

********* Begin Unverified *********

Theorem 8.1.1. Let L/K be the class field corresponding to N ⊂ J_{K}. Then a place v of K splits completely in L iff K_{v}^{×} ⊂ N.

Proof. See XI.3 of [6].

Let L/K be a finite abelian extension of number fields. For any place v of K let

U_{v} = \begin{cases}O_{K_{v}}^{×} & v finite \\ K_{v}^{×} & v infinite \end{cases}

Let I_{v} ⊂ Gal(L/K) denote the inertia group of some (any) prime lying over v. (This is welldefined because L/K is abelian.)

Theorem 8.1.2. Let L/K be the class field corresponding to N ⊂ J_{K}. Then (U_{v}, L/K) = I_{v}, in particular U_{v} ⊂ N iff v is unramified in L/K.

Proof. See XI.4 of [6].

********* End Unverified *********

8.2 Ray and ring class fields

8.3 Tables of class groups

********* Begin Unverified *********

8.3.1 Quadratic imaginary extensions

The following is a table of class groups of quadratic imaginary extensions of ℚ. It was constructed using unix sort, emacs, and the pari program

 for(d=1,100,if(issqrfree(k),
          k=buchgenfu(x^2+d);print(k[3,1][1],"&",k[5,1][2],"\\"),) )

See my section on pari for an implementation of issqrfree.

8.3. TABLES OF CLASS GROUPS 41

\begin{matrix}D_{K} & Cl_{K} \\ -3 & 1 \\ -4 & 1 \\ -7 & 1 \\ -8 & 1 \\ -11 & 1 \\ -15 & 2 \\ -19 & 1 \\ -20 & 2 \\ -23 & 3 \\ -24 & 2 \\ -31 & 3 \\ -35 & 2 \\ -39 & 4 \\ -40 & 2 \\ -43 & 1 \\ -47 & 5 \\ -51 & 2 \\ -52 & 2 \\ -55 & 4 \\ -56 & 4 \\ -59 & 3 \end{matrix} \begin{matrix}D_{K} & Cl_{K} \\ -67 & 1 \\ -68 & 4 \\ -71 & 7 \\ -79 & 5 \\ -83 & 3 \\ -84 & 2 × 2 \\ -87 & 6 \\ -88 & 2 \\ -91 & 2 \\ -95 & 8 \\ -103 & 5 \\ -104 & 6 \\ -107 & 3 \\ -115 & 2 \\ -116 & 6 \\ -119 & 10 \\ -120 & 2 × 2 \\ -123 & 2 \\ -127 & 5 \\ -131 & 5 \\ -132 & 2 × 2 \end{matrix} \begin{matrix}D_{K} & Cl_{K} \\ -136 & 4 \\ -139 & 3 \\ -143 & 10 \\ -148 & 2 \\ -151 & 7 \\ -152 & 6 \\ -155 & 4 \\ -159 & 10 \\ -163 & 1 \\ -164 & 8 \\ -167 & 11 \\ -168 & 2 × 2 \\ -179 & 5 \\ -183 & 8 \\ -184 & 4 \\ -187 & 2 \\ -191 & 13 \\ -195 & 2 × 2 \\ -199 & 9 \\ -203 & 4 \\ -211 & 3 \end{matrix} \begin{matrix}D_{K} & Cl_{K} \\ -212 & 6 \\ -228 & 2 × 2 \\ -232 & 2 \\ -244 & 6 \\ -247 & 6 \\ -248 & 8 \\ -255 & 6 × 2 \\ -259 & 4 \\ -260 & 4 × 2 \\ -263 & 13 \\ -264 & 4 × 2 \\ -267 & 2 \\ -271 & 11 \\ -276 & 4 × 2 \\ -280 & 2 × 2 \\ -283 & 3 \\ -287 & 14 \\ -291 & 4 \\ -292 & 4 \\ -295 & 8 \\ -296 & 10 \end{matrix}

Class groups of quadratic imaginary extensions of ℚ.

********* End Unverified *********

8.3.2 Gauss’ class number problem.

Gauss’ class number problem is, for each positive integer m, to determine the finitely many quadratic imaginary extensions of ℚ having class number m. This has been solved in principle by Goldfeld-Gross-Zagier. In a recent paper of Arno, Robinson and Wheeler [2] the problem is solved for all odd m ≤ 23. Here is a table of some of the known results. The ones with a * are conjectural; all others have been proved. (I computed this table using pari, and then verified that it was the same as the table given in [2].)

h  −D where D is a discriminant with class number h
1  3, 4, 7, 8, 11, 19, 43, 67, 163
2  15, 20, 24, 35, 40, 51, 52, 88, 91, 115, 123, 148, 187, 232, 267, 403, 427
3  23, 31, 59, 83, 107, 139, 211, 283, 307, 331, 379, 499, 547, 643, 883, 907
4  39, 55, 56, 68, 84, 120, 132, 136, 155, 168, 184, 195, 203, 219, 228, 259, 280, 291, 292,

312, 323, 328, 340, 355, 372, 388, 408, 435, 483, 520, 532, 555, 568, 595, 627, 667, 708, 715, 723, 760, 763, 772, 795, 955, 1003, 1012, 1027, 1227, 1243, 1387, 1411, 1435, 1507, 1555

5  47, 79, 103, 127, 131, 179, 227, 347, 443, 523, 571, 619, 683, 691, 739, 787, 947,

1051, 1123, 1723, 1747, 1867, 2203, 2347, 2683

6 *  87, 104, 116, 152, 212, 244, 247, 339, 411, 424, 436, 451, 472, 515, 628, 707, 771, 808, 835,

843, 856, 1048, 1059, 1099, 1108, 1147, 1192, 1203, 1219, 1267, 1315, 1347, 1363, 1432, 1563, 1588, 1603, 1843, 1915, 1963, 2227, 2283, 2443, 2515, 2563, 2787, 2923, 3235, 3427, 3523, 3763

7  71, 151, 223, 251, 463, 467, 487, 587, 811, 827, 859, 1163, 1171, 1483, 1523, 1627, 1787,

1987, 2011, 2083, 2179, 2251, 2467, 2707, 3019, 3067, 3187, 3907, 4603, 5107, 5923

8*  95, 111, 164, 183, 248, 260, 264, 276, 295, 299, 308, 371, 376, 395, 632, 651, 660, 712,

820, 840, 852, 868, 904, 915, 939, 952, 979, 987, 995, 1032, 1043, 1060, 1092, 1128, 1131, 1155, 1195, 1204, 1240, 1252, 1288, 1299, 1320, 1339, 1348, 1380, 1428, 1443, 1528, 1540, 1635, 1651, 1659, 1672, 1731, 1752, 1768, 1771, 1780, 1795, 1803, 1828, 1848, 1864, 1912, 1939, 1947, 1992, 1995, 2020, 2035, 2059, 2067, 2139, 2163, 2212, 2248, 2307, 2308, 2323, 2392, 2395, 2419, 2451, 2587, 2611, 2632, 2667, 2715, 2755, 2788, 2827, 2947, 2968, 2995, 3003, 3172, 3243, 3315, 3355, 3403, 3448, 3507, 3595, 3787, 3883, 3963, 4123, 4195, 4267, 4323, 4387, 4747, 4843, 4867, 5083, 5467, 5587, 5707, 5947, 6307

9  367, 419, 491, 563, 823, 1087, 1187, 1291, 1423, 1579, 199, 2003, 2803, 3163, 3259, 3307, 3547, 3643,

4027, 4243, 4363, 4483, 4723, 4987, 5443, 6043, 6427, 6763, 6883, 7723, 8563, 8803, 9067, 10627

11  167, 271, 659, 967, 1283, 1303, 1307, 1459, 1531, 1699, 2027, 2267, 2539, 2731, 2851, 2971, 3203,

3347, 3499, 3739, 3931, 4051, 5179, 5683, 6163, 6547, 7027, 7507, 7603, 7867, 8443, 9283, 9403, 9643, 9787, 10987, 13003, 13267, 14107, 14683, 15667

13  191, 263, 607, 631, 727, 1019, 1451, 1499, 1667, 1907, 2131, 2143, 2371, 2659, 2963, 3083, 3691, 4003, 4507, 4643,

5347, 5419, 5779, 6619, 7243, 7963, 9547, 9739, 11467, 11587, 11827, 11923, 12043, 14347, 15787, 16963, 20563

17  383, 991, 1091, 1571, 1663, 1783, 2531, 3323, 3947, 4339, 4447, 4547, 4651, 5483, 6203, 6379, 6451, 6827, 6907,

7883, 8539, 8731, 9883, 11251, 11443, 12907, 13627, 14083, 14779, 14947, 16699, 17827, 18307, 19963, 21067, 23563, 24907, 25243, 26083, 26107, 27763, 31627, 33427, 36523, 37123

19  311, 359, 919, 1063, 1543, 1831, 2099, 2339, 2459, 3343, 3463, 3467, 3607, 4019, 4139, 4327, 5059, 5147, 5527,

5659, 6803, 8419, 8923, 8971, 9619, 10891, 11299, 15091, 15331, 16363, 16747, 17011, 17299, 17539, 17683, 19507, 21187, 21211, 21283, 23203, 24763, 26227, 27043, 29803, 31123, 37507, 38707

21  431, 503, 743, 863, 1931, 2503, 2579, 2767, 2819, 3011, 3371, 4283, 4523, 4691, 5011, 5647, 5851, 5867, 6323,

6691, 7907, 8059, 8123, 8171, 8243, 8387, 8627, 8747, 9091, 9187, 9811, 9859, 10067, 10771, 11731, 12107, 12547, 13171, 13291, 13339, 13723, 14419, 14563, 15427, 16339, 16987, 17107, 17707, 17971, 18427, 18979, 19483, 19531, 19819, 20947, 21379, 22027, 22483, 22963, 23227, 23827, 25603, 26683, 27427, 28387, 28723, 28867, 31963, 32803, 34147, 34963, 35323, 36067, 36187, 39043, 40483, 44683, 46027, 49603, 51283, 52627, 55603, 58963, 59467, 61483

42 CHAPTER 8. CLASS FIELD THEORY

8.3.3 Real quadratic fields

********* Begin Unverified *********

The following is a table of class groups of real quadratic number fields.

\begin{matrix}D_{K} & Cl_{K} \\ 5 & 1 \\ 8 & 1 \\ 12 & 1 \\ 13 & 1 \\ 17 & 1 \\ 21 & 1 \\ 24 & 1 \\ 28 & 1 \\ 29 & 1 \\ 33 & 1 \\ 37 & 1 \\ 40 & 2 \\ 41 & 1 \\ 44 & 1 \\ 53 & 1 \\ 56 & 1 \\ 57 & 1 \\ 60 & 2 \\ 61 & 1 \\ 65 & 2 \\ 69 & 1 \\ 73 & 1 \\ 76 & 1 \end{matrix} \begin{matrix}D_{K} & Cl_{K} \\ 77 & 1 \\ 85 & 2 \\ 88 & 1 \\ 89 & 1 \\ 92 & 1 \\ 93 & 1 \\ 97 & 1 \\ 101 & 1 \\ 104 & 2 \\ 105 & 2 \\ 109 & 1 \\ 113 & 1 \\ 120 & 2 \\ 124 & 1 \\ 129 & 1 \\ 133 & 1 \\ 136 & 2 \\ 137 & 1 \\ 140 & 2 \\ 141 & 1 \\ 145 & 4 \\ 149 & 1 \\ 152 & 1 \end{matrix} \begin{matrix}D_{K} & Cl_{K} \\ 156 & 2 \\ 157 & 1 \\ 161 & 1 \\ 165 & 2 \\ 168 & 2 \\ 172 & 1 \\ 173 & 1 \\ 177 & 1 \\ 181 & 1 \\ 184 & 1 \\ 185 & 2 \\ 188 & 1 \\ 193 & 1 \\ 197 & 1 \\ 201 & 1 \\ 204 & 2 \\ 205 & 2 \\ 209 & 1 \\ 213 & 1 \\ 217 & 1 \\ 220 & 2 \\ 221 & 2 \\ 229 & 3 \end{matrix} \begin{matrix}D_{K} & Cl_{K} \\ 232 & 2 \\ 233 & 1 \\ 236 & 1 \\ 237 & 1 \\ 241 & 1 \\ 248 & 1 \\ 249 & 1 \\ 253 & 1 \\ 257 & 3 \\ 264 & 2 \\ 265 & 2 \\ 268 & 1 \\ 269 & 1 \\ 273 & 2 \\ 277 & 1 \\ 280 & 2 \\ 281 & 1 \\ 284 & 1 \\ 285 & 2 \\ 293 & 1 \\ 296 & 2 \\ 312 & 2 \\ 316 & 3 \end{matrix}

Class groups of real quadratic extensions of ℚ. ********* End Unverified *********

8.3.4 Cyclotomic fields

********* Begin Unverified ********* ********* End Unverified *********

Math 254B: Final Problem Set

********* Begin Unverified *********

Exercise 9.0.1 (1). a) Let K be a number field. Let E be the maximal abelian extension of K such that Spec(O_{E}) → Spec(O_{K}) is an étale map. Determine the group H ⊂ J_{K} to which E is the class field. b) Let h = #[Pic(O_{K})]. Show that h|#[Gal(E/K)] and that #[Gal(E/K)]/h is a power of 2.

Let

H = K^{×} ∏_{v∈S_{∞}}(K_{v}^{×})^{2} × ∏_{v∉S_{∞}} U_{v} ⊂ J_{K}.

By Theorem 8.1.2 H corresponds to the maximal extension of K which is unramified at all finite places v. For any finite extension L of K the map Spec(O_{L}) → Spec(O_{K}) is étale iff the corresponding extension of rings O_{L}/O_{K} is unramified. This is the case iff the extension L/K is unramified at all finite places. Thus H corresponds to E. Let

N = K^{×} ∏_{v∈S_{∞}} K_{v}^{×} × ∏_{v∉S_{∞}} U_{v}

be the norm subgroup corresponding to the Hilbert class field of K. Then h = [J_{K} : N] and H ⊂ N so

[E : K]/h = [J_{K} : H]/[J_{K} : N] = ([J_{K} : N][N : H])/[J_{K} : N] = [N : H]
= [K^{×} ∏_{v∈S_{∞}} K_{v}^{×} : K^{×} ∏_{v∈S_{∞}}(K_{v}^{×})^{2}] = [ ∏_{v∈S_{∞}} ℝ^{×} : ±1 ∏_{v∈S_{∞}} ℝ_{+}^{×}] = 2^{#[S_{∞}]−1}

Exercise 9.0.2 (2). Let K be a number field. Let K^{ab} be the maximal abelian extension of K and let K_{∞} = K(ζ_{m}, all m ∈ ℕ). Use class field theory to describe G = Gal(K^{ab}/K_{∞}).

Let ψ : C_{K} → Gal(K^{ab}/K) be the reciprocity map. Let N = {x ∈ C_{K} : ψ(x)|_{K_{∞}} = 1}. Then

N/ ker(ψ) ≅ ψ(N) ≅ G.

So to describe G it would suffice to describe N and ker(ψ). People don’t yet know how to describe ker(ψ). N is the intersection of the norm subgroups corresponding to the various K(ζ_{m}). It is tempting to say that this is the intersection of the inverse images of the corresponding subgroup of C_{ℚ} but I don’t know if this is true.

[Since K_{∞} contains the m-th roots of unity for all m, Kummer theory describes K^{ab}/K_{∞} in terms of K_{∞}^{×}/(K_{∞}^{×})^{m} for various m.] Exercise 9.0.3 (3). Let d ∈ ℤ^{×} be a square free (non-zero) integer. Use class field theory to determine the smallest m ∈ ℕ such that K = ℚ(\sqrt{d}) ⊂ ℚ(ζ_{m}).

Let D be the discriminant of K so K = ℚ(\sqrt{D}). The reciprocity map ψ_{K} : C_{ℚ} → Gal(K/ℚ) = {±1} factors as

C_{ℚ} ⟶ (( ⊕_{p|̸D} ℚ_{p}^{×}/ℤ_{p}^{×}) ⊕ ( ⊕_{p|D} ℚ_{p}^{×}) ⊕ (ℝ^{×}/ℝ_{+}^{×}))/ℚ^{×} → Gal(K/ℚ) = {±1}

We view ⊕_{p|̸D}ℚ_{p}^{×}/ℤ_{p}^{×} as I(D) = {n ∈ ℤ_{>0} : (n, D) = 1}. The map ψ_{K} on a factor ℤ_{p}^{×} with p|D is given by

ℤ_{p}^{×} → (ℤ/pℤ)^{×}/((ℤ/pℤ)^{×})^{2} ↪ ∏_{p|D}(ℤ/pℤ)^{×}/((ℤ/pℤ)^{×})^{2} \underrightarrow{∏} \{±1\}.

(If p = 2|D replace (ℤ/2ℤ)^{×}/((ℤ/2ℤ)^{×})^{2} by (ℤ/4ℤ)^{×}/((ℤ/4ℤ)^{×})^{2} in the above product.) Note also that (ℤ/pℤ)^{×}/((ℤ/pℤ)^{×})^{2} ≅ {±1} so the last map ∏ makes sense. For p|D, defining ψ_{K} on ℤ_{p}^{×} also defines ψ_{K} on ℚ_{p}^{×} since modulo ℚ^{×} every element of ⊕_{p|D}ℚ_{p}^{×} is in ⊕_{p|D}ℤ_{p}^{×}. Using the fact that ψ_{K}(ℚ^{×}) = 1 we that n ∈ I(D) maps to the image of n^{−1} under the map

∏_{p|D}(ℤ/pℤ)^{×}/((ℤ/pℤ)^{×})^{2} \underrightarrow{∏} \{±1\}.

We can use this description to answer problem 5 which asks for the kernel of ψ_{K}. When K is imaginary,

ker(ψ_{K}) = ℚ^{×} · {(n, ∏_{p|D} u_{p}, α) : n^{−1} ∏_{p|D} u_{p} ↦ α}

where α ∈ {±1} = ℝ^{×}/ℝ_{+}^{×} and u_{p} ∈ ℤ_{p}^{×}. When K is real

ker(ψ_{K}) = ℚ^{×} · {(n, ∏_{p|D} u_{p}) : n^{−1} ∏_{p|D} u_{p} ↦ 1}.

[This description leaves something to be desired.]

Since the same (finite) primes ramify in L = Q(ζ_{D}) as in K, the reciprocity map ψ_{L} factors as

C_{ℚ} ⟶ (( ⊕_{p|̸D} ℚ_{p}^{×}/ℤ_{p}^{×}) ⊕ ( ⊕_{p|D} ℚ_{p}^{×}) ⊕ (ℝ^{×}/ℝ_{+}^{×}))/ℚ^{×} → Gal(L/ℚ) = (ℤ/Dℤ)^{×}.

When p^{r}||D (exactly divides), the map ψ_{L} on ℤ_{p}^{×} is

ℤ_{p}^{×} → (ℤ/p^{r}ℤ)^{×} ⊂ (ℤ/Dℤ)^{×}.

From this explicit description we see that for any x ∈ C_{ℚ}, if ψ_{L}(x) = 1 then ψ_{K}(x) = 1. Thus N_{L/ℚ}(C_{L}) ⊂ N_{K/ℚ}(C_{K}) so K ⊂ L. Since L is the smallest cyclotomic field which is unramified outside D it must be the smallest which contains K.

Exercise 9.0.4 (4). Let m ∈ ℕ and let p be a rational prime. Determine the set of isomorphism classes of abelian extensions K of ℚ_{p} with [K : ℚ_{p}] = m.

By local class field theory this is the same as finding the index m subgroups of ℚ_{p}^{×}. Since an index m subgroup of ℚ_{p}^{×} necessarily contains (ℚ_{p}^{×})^{m} this is the same as finding the index m subgroups of ℚ_{p}^{×}/(ℚ_{p}^{×})^{m}. Since ℚ_{p}^{×} ≅ ℤ_{p}^{×} × ℤ we have

ℚ_{p}^{×}/(ℚ_{p}^{×})^{m} ≅ ℤ_{p}^{×}/(ℤ_{p}^{×})^{m} ⊕ (ℤ/mℤ).

The subgroup ℤ_{p}^{×}/(ℤ_{p}^{×})^{m} has index m and corresponds to the (unique) unramified abelian extension of ℚ_{p} of degree m. Let r be a positive integer such that p^{−r} < |m|_{p}^{2}. Then by Hensel’s lemma

ℤ_{p}^{×}/(ℤ_{p}^{×})^{m} ≅ (ℤ/p^{r}ℤ)^{×}/((ℤ/p^{r}ℤ)^{×})^{m}.

If p > 2 then (ℤ/p^{r}ℤ)^{×} is cyclic of order φ(p^{r}) = p^{r−1}(p − 1). If p = 2 and r > 1 then (ℤ/2^{r}ℤ)^{×} ≅ (ℤ/2ℤ) ⊕ (ℤ/2^{r−2}ℤ). Thus if m is prime to p − 1 and p, we see that ℚ_{p}^{×}/(ℚ_{p}^{×})^{m} ≅ ℤ/mℤ and so the degree m unramified extension is the only abelian extension of degree m.

As an example we exhibit all quadratic extensions of ℚ_{2}. We have

ℚ_{2}^{×}/(ℚ_{2}^{×})^{2} ≅ ℤ_{2}^{×}/(ℤ_{2}^{×})^{2} × 2ℤ/4ℤ ≅ \{\overline{1}, \overline{2}, \overline{3}, \overline{6}\}

where \overline{n} denotes the class of n in ℚ_{2}^{×}/(ℚ_{2}^{×})^{2}. The three quadratic extensions of ℚ_{2} are ℚ_{2}(\sqrt{−2}), ℚ_{2}(\sqrt{−3}), and ℚ_{2}(\sqrt{−6}) since their norm groups contain 2, 3, and 6, respectively.

45

Exercise 9.0.5 (5). If d is as in 3) determine the subgroup H ⊂ J_{ℚ} whose class field is ℚ(\sqrt{d}).

See the answer to problem 3. Exercise 9.0.6 (6a). Compute the class group of K = ℚ(ζ_{5}). By Theorem 3.4.1, D_{K} = 125. The Minkowski bound is

\frac{4!}{4^{4}} (\frac{4}{π} )^{2} · \sqrt{125} ≈ 1.7

which implies that K has trivial class group. Exercise 9.0.7 (6b). Compute the class group of K = ℚ(ζ_{7}). The Minkowski bound is

\frac{6!}{6^{6}} (\frac{4}{π} )^{3} · \sqrt{7^{5}} ≈ 4.1

Thus the class group is generated by the primes lying over 2 and 3. Since Frob(3) = 3 ∈ (ℤ/7ℤ)^{×} has order 6, we see that 3 remain s prime in O_{K} and so the primes over 3 do not contribute to the class group. Let α = 1 + ζ^{2} + ζ^{3} and β = 1 + ζ + ζ^{3}. Note that α and β are not units because α − 1 and β − 1 are both units. Their product is

αβ = 1 + ζ + ζ^{3} + ζ^{2} + ζ^{3} + ζ^{5} + ζ^{3} + ζ^{4} + ζ^{6} = 2ζ^{3}

so αβζ^{4} = 2. In terms of ideals this means that (αO_{K})(βO_{K}) = 2O_{K} so the primes lying over 2 are also all principal. Thus the class group is trivial. Exercise 9.0.8 (6c). Compute the Hilbert class field of K = ℚ(\sqrt{−15}). Since −15 ≡ 1 (mod 4), D_{K} = −15. Also N = [K : ℚ] = 2, r_{2} = 1, and O_{K} = ℤ[α] where α satisfies α^{2} + α + 4 = 0. By Theorem 3.3.1 every ideal class in Pic(O_{K}) contains an ideal of norm less than or equal to

\frac{2!}{2^{2}} (\frac{4}{π} ) · \sqrt{15} = \frac{2}{π} \sqrt{15} ≈ 2.47

Thus the Pic(O_{K}) is generated by the primes lying over 2ℤ. Since O_{K}/2O_{K} ≅ 𝔽_{2}[x]/(x(x + 1)) there are 2 distinct primes of O_{K} containing 2O_{K}. They are 𝔭 = (α, 2)O_{K} and \overline{𝔭} = (α+1, 2)O_{K}. If 𝔭 were principal then its generator would be an element a + bα ∈ O_{K} (a, b ∈ ℤ) such that

N_{K/ℚ}(a + bα) = (a + bα)(a + b\overline{α}) = a^{2} − ab + 4b^{2} = 2.

In order that the equation a^{2} − ba + 4b^{2} − 2 = 0 has a nonnegative integer solution in a, the discriminant b^{2} − 4(4b^{2} − 2) = −15b^{2} + 8 must be nonnegative. Since b ∈ ℤ this means that b = 0 so a^{2} = 2 which is impossible.

Because

𝔭^{2} = (α, 2)(α, 2) = (α^{2}, 2α, 4) = (α + 4, 2α, 4) = (α, 4) = (α)

we see that 𝔭 has order 2 in the class group so Pic(O_{K}) = ℤ/2ℤ

Let K_{1} = ℚ(\sqrt{−3}), K_{2} = ℚ(\sqrt{5}), and H = K_{1}.K_{2}. The primes 3 and 5 are the only primes which ramify in H/ℚ and they each have ramification index 2. This is because −3 ≡ 1 (mod 4) so D_{K_{1}} = −3 and 5 ≡ 1 (mod 4) so D_{K_{2}} = 5, so by Theorem 3.2.11 disc(H) = 15. [If 3 were totally ramified then it would ramify in each quadratic subfield of H, which is not the case.] Thus H/K is unramified at all finite primes and since H has no real places H/K is unramified at the infinite place. Thus H is the Hilbert class field of K. [The same argument with 3 and 5 replaced by primes p and q with p ≡ 3 (mod 4) and q ≡ 1 (mod 4) can be used to show that 2 divides the order of Pic(ℚ(\sqrt{−pq})) and hence that there are infinitely many quadratic imaginary extensions with nontrivial class group.]

I first tried to directly prove that 𝔭O_{H} is principal but was unsuccesful because 2 remains inert in each of the quadratic subfields of H. The trick is to move to another prime for which the computation is much easier. By exactly the same argument as above one sees that the prime 𝔮 of K lying over 3 is not principal. Since the class group of K has order 2, there must be an elements α ∈ O_{K} such that 𝔭 = (α)𝔮. Thus to show that 𝔭 becomes principal in O_{H} is equivalent to showing that 𝔮 becomes principal in O_{H}. Since 5 has no square root in 𝔽_{3}, the prime 𝔔 of H lying over 3 has degree 2. Thus to show that 𝔔 is principal it suffices to exhibit an element of O_{H} having norm 9. One checks immediately that \sqrt{−3} is such an element.

Exercise 9.0.9 (6d). Compute the Hilbert class field of K = ℚ(\sqrt{−20}) = ℚ(\sqrt{−5}). The Minkowski bound is

\frac{2!}{2^{2}} (\frac{4}{π} )^{1} · \sqrt{20} ≈ 2.8

so the class group is generated by the prime 𝔭 lying over 2. Since 𝔭^{2} = 2O_{K} is principal the class group is either trivial or ℤ/2ℤ. The ideal 𝔭 is not principal iff there is no element of O_{K} whose norm is 2. Since O_{K} = ℤ[\sqrt{−5}] this reduces to checking whether or not a^{2} − 5b^{2} = 2 has a solution with a, b ∈ ℤ. Since 2 is not a square modulo 5 it follows that 𝔭 is not principal so Pic(K) = ℤ/2ℤ.

The Hilbert class field is H = ℚ(\sqrt{−1}, \sqrt{−5}). Since 2 and 5 are the only primes which can ramify in ℚ(\sqrt{−1}) or K it follows that H is unramified outside 2 and 5. Since 2 and 5 both ramify in K, to show that H is unramified over K it suffices to show that neither 2 nor 5 is totally ramified. This follows because 2 does not ramify in the subfield ℚ(\sqrt{5}) ⊂ H and 5 does not ramify in the subfield ℚ(\sqrt{−1}) ⊂ H.

To check that the prime 𝔭 of K lying over 2 becomes principal in H it suffices to show that there is an element α ∈ O_{H} such that N_{H/ℚ}(α) = 2^{2}. [This is enough because the prime lying over 2 has degree 2 in H/ℚ, and so such an element α must have prime factorization equal to the unique prime of H lying over 2.] Such an element is α = 1 + \sqrt{−1} since

N_{H/ℚ}(α) = N_{ℚ(\sqrt{−1})/ℚ}(N_{H/ℚ(\sqrt{−1})}(1 + \sqrt{−1})) = N_{ℚ(\sqrt{−1})/ℚ}((1 + \sqrt{−1})^{2}) = (−2)^{2} = 4.

Exercise 9.0.10 (6e). Compute the Hilbert class field of K = ℚ(\sqrt{−23}) The Minkowski bound is

\frac{2!}{2^{2}} (\frac{4}{π} )^{1} · \sqrt{23} ≈ 3.05

Let α satisfy α^{2} − α + 6 = 0 so O_{K} = ℤ[α]. Let 𝔭_{1} and 𝔭_{2} denote the primes of O_{K} lying over 2 so 𝔭_{1} = (2, α) and 𝔭_{2} = (2, α − 1). Let 𝔮_{1} = (3, 2α) and 𝔮_{2} denote the primes lying over 3. First note that 𝔭_{1}𝔭_{2} = (2) is principal so [𝔭_{1}]^{−1} = [𝔭_{2}] in the class group. Second note that

𝔭_{1}𝔮_{1} = (2, α)(3, 2α) = (6, 4α, 3α, 2α^{2}) = (6, α) = (α)

so [𝔮_{1}] = [𝔭_{1}]^{−1} = [𝔭_{2}]. Thus the class group is generated by [𝔭_{1}]. Since the norm form a^{2} + ab + 6b^{2} doesn’t represent 2, we see that 𝔭_{1} is not principal. In the class group, 𝔭_{1}^{2} is equivalent to either 𝔭_{2} or the unit ideal. Since 𝔭_{2} is the inverse of 𝔭_{1}, we see that the class group has order either 2 or 3. I could proceed at this point by showing that 𝔭_{1}^{2} is not principal and hence the class group is ℤ/3ℤ. Instead, I will apply the class field theory. Let H be the splitting field of the cubic x^{3} − x + 1. Then H is an unramified abelian extension of K of degree 3 hence the class group has order 3. Exercise 9.0.11 (6f). The class group of K = ℚ(\sqrt{3}) is trivial. The Minkowski bound is

\frac{2!}{2^{2}} (\frac{4}{π} )^{0} · \sqrt{12} ≈ \sqrt{3}

so the class group is generated by the (unique) prime 𝔭 lying over 2. But

N_{K/ℚ}(1 + \sqrt{3}) = (1 + \sqrt{3})(1 − \sqrt{3}) = −2

so 𝔭 is principal, generated by 1 + \sqrt{3}. [Use the fact that 1 + \sqrt{3} is an integer and consider its factorization as a product of prime ideals.] Exercise 9.0.12 (7). Verify the principal ideal theorem in the cases of problem 6.

This is done in the solutions to problem 6. Exercise 9.0.13 (8). Verify the theorem from page 224 of Lang’s book in the cases k = ℚ_{11} and K is an abelian extension of k of degree 12.

A degree 12 extension of ℚ_{11} can have no wild ramification since 11 does not divide 12. Thus by Theorem 1 in section 8 of Fröhlich’s article Local Fields in [4] we see that all higher ramification groups Γ_{i}, i ≥ 1, must be 0. By Theorem 8.1.2 we know that the theorem on page 224 holds for Γ_{0}. [For fun, I wrote down the three abelian extensions K of k of degree 12 – they are in my notes.]

47

********* End Unverified *********

chapterGenus One Curves of Arbitrary Index

9.1 Indexes of Curves

Let X be a (nonsingular) curve of genus g over a number field K. Let

Div_{K}(X) = Div(X)^{Gal(K̄/K)}

denote the group of K rational divisors on X. The map which assigns to a divisor its degree is a homomorphism

deg : Div_{K}(X) → ℤ

whose image is a subgroup mℤ of ℤ. (The degree of a prime divisor P is the degree of the smallest extension of K over which P is rational, i.e., the field cut out by the subgroup of Gal(K̄/K) which fixes P.) We say that the integer m is the index of X. The canonical divisor is a rational divisor of degree 2g − 2 so the index of X is bounded by (and in fact divides) 2g − 2. When g = 1 this condition does not restrict m. In [Lang-Tate, 1958] it is proved that the index is completely arbitrary, a fact conjectured by Artin but unknown until then. In their construction the base field K is not under control. They say,

“... given any positive integer m one can construct a function field in one variable of genus 1, over a suitable algebraic number field K, the degrees of whose divisors rational over K are exactly the multiples of m. One still does not have such examples when K = ℚ.”

Galois cohomology of abelian varieties provides a powerful tool in the search for an answer to this question. If X is a genus one curve over K then the index of X is the same as the index of any of the cohomology classes c ∈ H^{1}(K, E) corresponding to X. Here E is the Jacobian variety of X. Since X has genus one, E is an elliptic curve. The index of a cohomology class c is defined to be the greatest common divisor of the degrees of the extensions of K in which c restricts to zero.

9.2 Euler Systems and Indexes of Cohomology Classes

9.2.1 Introduction

Euler Systems

Let E/ℚ be a modular elliptic curve with conductor N. Fix a surjective morphism π : X_{0}(N) → E. Let L(E, s) denote the L-series associated to E. Assume that L(E, 1) ≠ 0. Let K be a quadratic imaginary extension of ℚ with discriminant D_{K} in which all primes dividing N split. Fix an integral ideal N such that O_{K}/ N ≅ ℤ/Nℤ. Let H be the Hilbert class field of K. Let x_{H} be a point on X_{0}(N)(H) corresponding to (ℂ/O_{K}, N^{−1}/O_{K}). Let y_{H} = π(x_{H}), y_{K} = Tr_{H/K}(y_{H}), and y = y_{K} − y_{K}^{τ}, where τ denotes complex conjugation. By recent work of Bump-Friedberg-Hoffstein we know that K can be chosen so that y has infinite order. In this case Kolyvagin has proven that E(K) has rank 1, that E(ℚ) has rank 0, and that � (E/ℚ) is finite. This is accomplished in three steps. First, using the theory of modular curves and complex multiplication one constructs an “Euler System” of points y_{ℓ} ∈ E(\overline{ℚ}). The second step is to take the “derivative” of this Euler system to obtain a collection of cohomology classes c_{ℓ,p^{n}} ∈ H^{1}(ℚ, E). The third step is to study the remarkable properties of these cohomology classes and use them to bound the orders of certain Selmer groups.

The Result

Using the classes c_{ℓ,p^{n}} of Kolyvagin’s Euler system I can prove the following.

9.2. EULER SYSTEMS AND INDEXES OF COHOMOLOGY CLASSES 49

Theorem 9.2.1. Let E be a modular curve with L(E, 1) ≠ 0 as above. Then there is a finite set S of primes such that for all positive integers m not divisible by any prime in S there are infinitely many cohomology classes having order and index both equal to m. Corollary 9.2.2. There is a finite set S of primes with the following property. Let K be a number field and m a positive integer not divisible by any prime in S. Then there are infinitely many genus one curves X over K of index m.

In terms of the Euler system we actually prove the following. Theorem 9.2.3. For all but finitely many primes p and for each n ≥ 1 there are infinitely many primes ℓ such that the cohomology classes c_{ℓ,p^{n}} have order and index both equal to p^{n}.

9.2.2 Background

Fix an embedding \overline{ℚ} ↪ ℂ and write τ for the complex conjugation on \overline{ℚ}. Rubin’s Paper We require the following results from Rubin’s paper. Assume p > 2 and that D_{K} ≠ 3, 4. Lemma 9.2.4. Fix a prime p. Suppose ℓ is a prime not dividing pD_{K}N, r >0, and Frob_{ℓ}(K(E_{p^{r}})/ℚ) = [τ]. Then if Ẽ denotes the reduction of E modulo ℓ and a_{ℓ} = ℓ + 1 − #[Ẽ(𝔽_{ℓ})],

(i) p^{r}|a_{ℓ} and p^{r}|ℓ + 1, (ii) ℓ remains prime in K, (iii) E(ℚ_{ℓ})_{p^{r}} ≅ Ẽ(𝔽_{ℓ})_{p^{r}} ≅ ℤ/p^{r}ℤ, (E(K_{ℓ})_{p^{r}})^{−} ≅ (Ẽ(𝔽_{ℓ^{2}})_{p^{r}})^{−}) ≅ ℤ/p^{r}ℤ.

For each place v of ℚ let m_{v} = [H^{1}(ℚ_{v}^{unr}/ℚ_{v}, E(ℚ_{v}^{unr}))]. Each m_{v} is finite and all but finitely many are zero, so m(p) = sup{ord_{p}(m_{v}) : all places v of ℚ} is a well-defined integer, equal to zero for almost all p. Proposition 9.2.5. Suppose ℓ is a prime not dividing pD_{K}N and Frob_{ℓ}(K(E_{p^{r}})/ℚ) = [τ], where r = n + m(p). Then there is an element c_{ℓ,p^{n}} ∈ H^{1}(ℚ, E)_{p^{n}} such that

• res_{v}(c_{ℓ,p^{n}}) = 0 for all v ≠ ℓ,

• the order of res_{ℓ}(c_{ℓ,p^{n}}) in H^{1}(ℚ_{ℓ}, E)_{p^{n}} is equal to the order of y in E(K_{ℓ})/p^{n}E(K_{ℓ}),

• if p |̸[H : K] then c_{ℓ,p^{n}} has index dividing p^{n}.

For each prime p and positive integer n let b(p^{n}) = a_{1} + a_{2} where p^{a_{1}} is the order of the largest ℚ-rational cyclic subgroup of E_{p^{∞}} and p^{a_{2}} is the exponent of H^{1}(K(E_{p^{n+m(p)}})/K, E_{p^{n}}). Theorem 9.2.6. For all but finitely many primes p the numbers b(p^{n}) are zero for all n ≥ 1.

For any t ∈ H^{1}(K, E_{p^{n}}), write t̂ for the image of t under the restriction map

res : H^{1}(K, E_{p^{n}}) → Hom(Gal(K̄/K(E_{p^{n+m(p)}})), E_{p^{n}}).

Lemma 9.2.7. Suppose t ∈ H^{1}(K, E_{p^{n}})^{±} and the image of the homomorphism t̂ is a cyclic subgroup of E_{p^{n}}. Then the order of t is at most p^{b(p^{n})}. Other Results Theorem 9.2.8. Assume that the point y_{K} has infinite order in E(K). Then the group E(K) has rank 1.

[See Gross’ theorem 1.3.]

9.2.3 A Result About the Euler System

Lemma 9.2.9. Let F/L be a Galois extension of fields, let A be a (discrete) Gal(L̄/L)-module, and let φ be a Gal(L̄/L)-module homomorphism such that the sequence

0 → A_{φ} → A ⟶^{φ} A → 0

is exact (i.e., φ is surjective). Then the map δ : H^{0}(F, A) → H^{1}(F, A_{φ}) is Gal(F/L)-equivariant.

Proof. For x ∈ H^{0}(F, A) let y ∈ A be such that φ(y) = x. Let f : Gal(L̄/L) → A_{φ} be the cocycle defined by f(σ) = σy − y, so [f] = δ(x). Then for any γ ∈ Gal(F/L) and σ ∈ Gal(L̄/L),

γ ∗ f(σ) = γf(γ^{−1}σγ) = γ(γ^{−1}σγy − y) = σ(γy) − γy.

Since φγy = γφy = γx, it follows that [γ ∗ f] = δ(γx).

Let G_{ℚ} = Gal(\overline{ℚ}/ℚ).

Lemma 9.2.10. Let E/ℚ be an elliptic curve, let φ : E → E be a ℚ-rational isogeny, and let H ⊂ E(\overline{ℚ}) be a G_{ℚ}-invariant subgroup. Then φ^{−1}(H) is G_{ℚ}-invariant.

Proof. View E(\overline{ℚ}) as a module over the ring ℤ[G_{ℚ}] and use the fact that the inverse image of a submodule is a submodule. [If x ∈ φ^{−1}(H) and σ ∈ G_{ℚ} then φ(σx) = σφ(x) ∈ σH = H so σx in φ^{−1}(H).]

Let K([p^{n}]^{−1}w) denote the field obtained from K by adjoining the coordinates of all p^{n}-th roots of w. This can also be thought of as the smallest extension of K over which all p^{n}-th roots of w are defined. In general this field may not be Galois over ℚ, but when w ∈ E(K)^{−} it is.

Corollary 9.2.11. If w ∈ E(K)^{−} then K([p^{n}]^{−1}w) is Galois over ℚ.

Proof. Since G_{ℚ} acts on w by ±1, the subgroup ℤw is G_{ℚ}-invariant. Thus [p^{n}]^{−1}ℤw is G_{ℚ}- invariant. Now use the fact that the group operations on E are defined over ℚ and so the field of definition of a set of points is the same as the field of definition of all of the points in the subgroup they generate.

Proposition 9.2.12. Suppose w ∈ E(K) satisfies the following two conditions.

(i) τw = ±w

(ii) the image of w in E(K)/p^{n}E(K) has order strictly greater than p^{b(p^{n})}

Then there are infinitely many primes ℓ such that Frob_{ℓ}(K(E_{p^{n+m(p)}})/ℚ) = [τ] and the image of w in E(K_{ℓ})/p^{n}E(K_{ℓ}) is nonzero.

Proof. We shall work with the exact sequence

0 → p^{n}E(K) ↪ E(K) ⟶^{δ} H^{1}(K, E_{p^{n}}) ⟶^{res} Hom(Gal(K̄/K(E_{p^{n+m(p)}}), E_{p^{n}}))

Suppose w ∈ E(K)^{−}, the argument when w ∈ E(K)^{+} being identical. Then δ(w) ∈ H^{1}(K, E_{p^{n}})^{−}. Since p^{b(p^{n})}δ(w) ≠ 0, δ(w) ∉ ker(res) so

0 ≠ res(δ(w)) = { \atop δ(w)} ∈ Hom(Gal(K̄/K(E_{p^{n+m(p)}}), E_{p^{n}}).

Fix x so that p^{n}x = w. Then δ(w) is represented by the cocycle σ ↦ σ(x) − x so { \atop δ(w)} is the homomorphism { \atop δ(w)}(σ) = σ(x) − x. The minimal splitting field (i.e., fixed field of the kernel of

{ \atop δ(w)}) is thus

L = K(E_{p^{n+m(p)}})(x) = K([p^{n}]^{−1}w).ℚ(E_{p^{n+m(p)}})

which is Galois over ℚ [note that any two p^{n}-th roots of w differ by an element of E_{p^{n}}.] Thus we have a tower of Galois extensions of ℚ:

ℚ −K −K(E_{p^{n+m(p)}}) −L.

Let σ ∈ Gal(L/K(E_{p^{n+m(p)}})). If ℓ is such that Frob_{ℓ}(L/ℚ) = [στ] then Frob_{ℓ}(K(E_{p^{n+m(p)}})/ℚ) = [τ]. Then fFor any prime λ of L lying over ℓ, the degree [L_{λ} : ℚ] equals the order of στ. Completing each field at ℓ we obtain the exact sequence

0 → p^{n}E(K_{ℓ}) ↪ E(K_{ℓ}) ⟶^{δ_{ℓ}} H^{1}(K_{ℓ}, E_{p^{n}}) ⟶^{res} Hom(Gal(K̄_{ℓ}/K_{ℓ}(E_{p^{n+m(p)}})), E_{p^{n}}).

The minimal splitting field of { \atop δ_{ℓ}(w)} is K_{ℓ}(E_{p^{n+m(p)}})(z) where z is such that p^{n}z = w. Since

L/ℚ is Galois, L_{λ} = K_{ℓ}(E_{p^{n+m(p)}})(z). Because [K_{ℓ}(E_{p^{n+m(p)}}) : ℚ_{ℓ}] = 2, { \atop δ_{ℓ}(w)} = 0 if and only if [L_{λ} : ℚ_{ℓ}] = 2 which is true if and only if (στ)^{2} = 1.

If (στ)^{2} = 1 for all σ ∈ Gal(L/K(E_{p^{n+m(p)}})) then since τ ∗ { \atop δ(w)} = −{ \atop δ(w)},

0 = { \atop δ(w)}((στ)^{2}) = { \atop δ(w)}(στστ) = { \atop δ(w)}(σ) + { \atop δ(w)}(τ^{−1}στ) = { \atop δ(w)}(σ) − τ{ \atop δ(w)}(σ).

Thus { \atop δ(w)}(σ) = τ{ \atop δ(w)}(σ), i.e., the image of the homomorphism { \atop δ(w)} is contained in E_{p^{n}}^{+}. Since

E_{p^{n}}^{+} is cyclic, the image of { \atop δ(w)} must also be cyclic. By an above lemma this implies that δ(w) is annihilated by p^{b(p^{n})} contrary to our hypothesis [since the image of w was assumed to have order greater than p^{b(p^{n})} in E(K)/p^{n}E(K) and this order is the same as the order of δ(w).] Therefore there must be a σ ∈ Gal(L/K(E_{p^{n+m(p)}})) such that στ has order different than 2. For such a σ and any ℓ such that Frob_{ℓ}(L/ℚ) = [στ] we have shown that δ_{ℓ}(w) ≠ 0 so the image of w in E(K_{ℓ})/p^{n}E(K_{ℓ}) is not zero.

The above proposition can be extended to obtain more precise information about the local order of w.

Corollary 9.2.13. Let w be as in Proposition 9. Then there exists infinitely many primes ℓ such that Frob_{ℓ}(K(E_{p^{n+m(p)}})/ℚ) = [τ] and the image of w in E(K_{ℓ})/p^{n}E(K_{ℓ}) has order at least as large as the order of the image of p^{b(p^{n})}w in E(K)/p^{n}E(K).

Proof. Let a be the integer such that p^{a}p^{b(p^{n})}w ≠ 0 in E(K)/p^{n}E(K) but p^{a+1}p^{b(p^{n})}w = 0 in E(K)/p^{n}E(K). Thus p^{a+1} is the order of the p^{b(p^{n})}w in E(K)/p^{n}E(K). Let w′ = p^{a}w, so w′ is not annihilated by p^{b(p^{n})} in E(K)/p^{n}E(K). By the Proposition there are infinitely many primes ℓ such that Frob_{ℓ}(K(E_{p^{n+m(p)}})/ℚ) = [τ] and such that the image of w′ in E(K_{ℓ})/p^{n}E(K_{ℓ}) is not zero. For such ℓ, it follows that w has order at least p^{a+1} in E(K_{ℓ})/p^{n}E(K_{ℓ}), as required.

9.2.4 Proofs of Theorems 1 and 3

Proof of theorem 3. Fix an elliptic curve E/ℚ and an imaginary quadratic field K as in the introduction. Assume that y_{K} has infinite order. Define the set S of primes as follows

S = {p : b(p^{n}) ≠ 0 for some n ≥ 1} ∪ {p : y ∈ pE(K)}
∪{p : p|[E(K)_{tor}]} ∪ {p : p|[H : K]}

Fix a prime p ∉ S. Since E(K) has rank one the image of y in E(K)/p^{n}E(K) has order p^{n}. By Corollary [?] there are infinitely many primes ℓ such that Frob_{ℓ}(K(E_{p^{n}})/ℚ) = [τ] and the image of y in E(K_{ℓ})/p^{n}E(K_{ℓ}) has order p^{n}. By proposition [??] the cohomology classes c_{ℓ,p^{n}} have order and index both equal to p^{n}.

Theorem 9.2.14. Let S be the set of primes in the above proof and let L/ℚ be a number field. Then for all primes p ∉ S and all integers n ≥ 1, there exists infinitely many elements of H^{1}(L, E) having order and index both equal to p^{n}.

Proof. Fix p ∉ S and n ≥ 1. Let L′ denote the Galois closure of L/ℚ. Let ℓ be one of the infinitely many primes such that c_{ℓ,p^{n}} ∈ H^{1}(ℚ, E) has order and index equal to p^{n} and such that ℓ does not ramify in L′/ℚ. Let

res_{L} : H^{1}(ℚ, E) → H^{1}(L, E)

be the restriction map. If p^{n−1} res_{L}(c_{ℓ,p^{n}}) = 0 then p^{n−1}c_{ℓ,p^{n}} ∈ H^{1}(L′/ℚ, E). But L′/ℚ is unramified at ℓ, ℓ is a prime of good reduction for E, and ℓ ≠ p so we conclude that locally at ℓ, p^{n−1}c_{ℓ,p^{n}} = 0 (this is a result proved in [Lang-Tate, 1958]). This is impossible because of the second condition in Proposition [?].

9.3 Order and index questions for H^{1}(ℚ, E)

9.3.1 Cohomological criterion for order=index

9.3.2 Order=index for elements of � (E/ℚ)

9.3.3 Cassels’ example that order need not equal index

Introduction

In the paper [3] Cassels provides a counterexample which shows that the order need not equal the index for elements of H^{1}(ℚ, E). This is in contrast to the local case in which the order equals the index, see Lichtenbaum [7].

Cassels shows that for a certain elliptic curve E there are infinitely many elements of H^{1}(ℚ, E)[2] having index 4. In this section we work through his proof for a specific cohomology class.

Fix α, β ∈ \overline{ℚ} such that α^{2} = 3 and β^{2} = −11, and let K = ℚ(α, β). The galois group Gal(K/ℚ) is generated by σ_{α} and σ_{β} where σ_{α}(α) = −α, σ_{α}(β) = β, σ_{β}(α) = α and σ_{β}(β) = −β. Let c ∈ H^{1}(K/ℚ, E) ⊂ H^{1}(ℚ, E) be the class defined by the cocycle which sends σ_{α} to (0, 0) and σ_{β} to (−1, 0). Since (0, 0) and (−1, 0) are 2-torsion points, c has order 2. We show that c has index 4.

Properties of E

The torsor corresponding to c

Period-index criterion

Concluding remarks

9.3.4 Order=index over a local field

9.4 The basic exact sequence

Let X be a curve over a field k. Let k̄ denote the separable closure of k, and let K̄ denote the function field of X̄ = X ×_{k} k̄. For any Gal(k̄/k)-module A let H^{i}(A) = H^{i}(Gal(k̄/k), A). We define^{1} the Brauer group of X,

Br(X) = ker(H^{2}(K̄^{×}) → H^{2}(Div(X̄)))

where the map is induced by the map which sends an element of K̄^{×} to its divisor.

Theorem 9.4.1. There is a natural exact sequence

0 → Pic(X) → H^{0}(Pic(X̄)) → Br(k) → Br(X) → H^{1}(Pic(X̄)) → H^{3}(k̄^{×})

1It is proved in the appendix to [7] that Br(X) = H2_{ét}(X, 𝔾_{m}).

To prove the theorem we require two vanishing lemmas. The first is essentially Shapiro’s Lemma and the second is a generalization of Hilbert’s Theorem 90.

Lemma 9.4.2. H^{1}(Div(X̄)) = 0.

Proof. Let n be any positive integer. Since H^{1}(Div(X̄)) is a torsion group it suffices to show that H^{1}(Div(X̄))[n] = 0. Let A be the cokernel of multiplication by n on Div(X̄) so that the sequence

0 → Div(X̄) ⟶^{n} Div(X̄) → A → 0

is exact. We obtain the cohomology exact sequence

H^{0}(Div(X̄)) → H^{0}(A) → H^{1}(Div(X̄)) ⟶^{n} H^{1}(Div(X̄)).

To show that H^{1}(Div(X̄))[n] = 0 it suffices to show that the map H^{0}(Div(X̄)) → H^{0}(A) is surjective. This is the case because

A = Div(X̄)/n Div(X̄) ≅ {∑ m_{i}P_{i} : m_{i} ∈ ℤ/nℤ, P_{i} ∈ X(k̄)}

and the Galois action commutes with the ℤ/nℤ-coefficients.

Lemma 9.4.3. H^{1}(K̄^{×}) = 0.

Proof. ?

We can now prove the theorem.

Proof of Theorem. The exact sequence

0 → K̄^{×}/k̄^{×} → Div(X̄) → Pic(X̄) → 0

gives the exact sequence

H^{0}(Div(X̄)) → H^{0}(Pic(X̄)) → H^{1}(K̄^{×}/k̄^{×}) → H^{1}(Div(X̄)) = 0.

From the exact sequence

0 → k̄^{×} → K̄^{×} → K̄^{×}/k̄^{×} → 0

we obtain the exact sequence

0 = H^{1}(K̄^{×}) → H^{1}(K̄^{×}/k̄^{×}) → Br(k) → H^{2}(K̄^{×}) → H^{2}(K̄^{×}/k̄^{×}) → H^{3}(k̄^{×}).

Since the divisor of an element of k̄^{×} is zero, the image of Br(k) in H^{2}(K̄^{×}) lies in Br(X). The image of H^{0}(Div(X̄)) = Div(X) in H^{0}(Pic(X̄)) is Pic(X). We thus have the following commutative diagram with exact rows and columns:

0  0
↓  ↓
Br(k) →  Br(X)  H^{1}(Pic(X̄))
||  ↓  ↓
Br(k) → H^{2}(K̄^{×}) → H^{2}(K̄^{×}/k̄^{×}) → H^{3}(k̄^{×})
↓  ↓
H^{2}(Div(X̄)) = H^{2}(Div(X̄)).

The map H^{2}(K̄^{×}) → H^{2}(K̄^{×}/k̄^{×}) induces a map Br(X) → H^{1}(Pic(X̄)). Combining all of these facts yields the basic exact sequence.

Chapter 10

54 10.1. TABLES 55

Elliptic Curves

10.1 Tables

10.1.1 Cremona’s table for N ≤ 64

curve  model  r  t
11A1  0  -1  1  -10  -20  0  5
11A2  0  -1  1  -7820  -263580  0  1
11A3  0  -1  1  0  0  0  5
14A1  1  0  1  4  -6  0  6
14A2  1  0  1  -36  -70  0  6
14A3  1  0  1  -171  -874  0  2
14A4  1  0  1  -1  0  0  6
14A5  1  0  1  -2731  -55146  0  2
14A6  1  0  1  -11  12  0  6
15A1  1  1  1  -10  -10  0  4 × 2
15A2  1  1  1  -135  -660  0  2 × 2
15A3  1  1  1  -5  2  0  4 × 2
15A4  1  1  1  35  -28  0  8
15A5  1  1  1  -2160  -39540  0  2
15A6  1  1  1  -110  -880  0  2
15A7  1  1  1  -80  242  0  4
15A8  1  1  1  0  0  0  4
17A1  1  -1  1  -1  -14  0  4
17A2  1  -1  1  -6  -4  0  2 × 2
17A3  1  -1  1  -91  -310  0  2
17A4  1  -1  1  -1  0  0  4
19A1  0  1  1  -9  -15  0  3
19A2  0  1  1  -769  -8470  0  1
19A3  0  1  1  1  0  0  3
20A1  0  1  0  4  4  0  6
20A2  0  1  0  -1  0  0  6
20A3  0  1  0  -36  -140  0  2
20A4  0  1  0  -41  -116  0  2
21A1  1  0  0  -4  -1  0  4 × 2
21A2  1  0  0  -49  -136  0  2 × 2
21A3  1  0  0  -39  90  0  8
21A4  1  0  0  1  0  0  4
21A5  1  0  0  -784  -8515  0  2
21A6  1  0  0  -34  -217  0  2
24A1  0  -1  0  -4  4  0  4 × 2
24A2  0  -1  0  -24  -36  0  2 × 2
24A3  0  -1  0  -64  220  0  4
24A4  0  -1  0  1  0  0  4
24A5  0  -1  0  -384  -2772  0  2
24A6  0  -1  0  16  -180  0  2
26A1  1  0  1  -5  -8  0  3
26A2  1  0  1  -460  -3830  0  1
26A3  1  0  1  0  0  0  3
26B1  1  -1  1  -3  3  0  7
26B2  1  -1  1  -213  -1257  0  1
27A1  0  0  1  0  -7  0  3
27A2  0  0  1  -270  -1708  0  1
27A3  0  0  1  0  0  0  3
27A4  0  0  1  -30  63  0  3
30A1  1  0  1  1  2  0  6
30A2  1  0  1  -19  26  0  6 × 2
30A3  1  0  1  -14  -64  0  2
30A4  1  0  1  -69  -194  0  6
30A5  1  0  1  -289  1862  0  6
30A6  1  0  1  -334  -2368  0  2 × 2
30A7  1  0  1  -5334  -150368  0  2
30A8  1  0  1  -454  -544  0  2
32A1  0  0  0  4  0  0  4
32A2  0  0  0  -1  0  0  2 × 2
32A3  0  0  0  -11  -14  0  2
32A4  0  0  0  -11  14  0  4
33A1  1  1  0  -11  0  0  2 × 2
33A2  1  1  0  -6  -9  0  2
33A3  1  1  0  -146  621  0  4
33A4  1  1  0  44  55  0  2
34A1  1  0  0  -3  1  0  6
34A2  1  0  0  -43  105  0  6
34A3  1  0  0  -103  -411  0  2
34A4  1  0  0  -113  -329  0  2
35A1  0  1  1  9  1  0  3
35A2  0  1  1  -131  -650  0  1
35A3  0  1  1  -1  0  0  3
36A1  0  0  0  0  1  0  6
36A2  0  0  0  -15  22  0  6
36A3  0  0  0  0  -27  0  2
36A4  0  0  0  -135  -594  0  2
37A1  0  0  1  -1  0  1  1
37B1  0  1  1  -23  -50  0  3
37B2  0  1  1  -1873  -31833  0  1
37B3  0  1  1  -3  1  0  3
38A1  1  0  1  9  90  0  3
38A2  1  0  1  -86  -2456  0  1
38A3  1  0  1  -16  22  0  3
38B1  1  1  1  0  1  0  5
38B2  1  1  1  -70  -279  0  1
39A1  1  1  0  -4  -5  0  2 × 2
39A2  1  1  0  -69  -252  0  2
39A3  1  1  0  -19  22  0  4

curve  model  r  t
39A4  1  1  0  1  0  0  2
40A1  0  0  0  -7  -6  0  2 × 2
40A2  0  0  0  -107  -426  0  2
40A3  0  0  0  -2  1  0  4
40A4  0  0  0  13  -34  0  4
42A1  1  1  1  -4  5  0  8
42A2  1  1  1  -84  261  0  4 × 2
42A3  1  1  1  -104  101  0  2 × 2
42A4  1  1  1  -1344  18405  0  4
42A5  1  1  1  -914  -10915  0  2
42A6  1  1  1  386  1277  0  2
43A1  0  1  1  0  0  1  1
44A1  0  1  0  3  -1  0  3
44A2  0  1  0  -77  -289  0  1
45A1  1  -1  0  0  -5  0  2
45A2  1  -1  0  -45  -104  0  2 × 2
45A3  1  -1  0  -720  -7259  0  2
45A4  1  -1  0  -90  175  0  2 × 2
45A5  1  -1  0  -1215  16600  0  2 × 2
45A6  1  -1  0  315  1066  0  2
45A7  1  -1  0  -19440  1048135  0  2
45A8  1  -1  0  -990  22765  0  2
46A1  1  -1  0  -10  -12  0  2
46A2  1  -1  0  -170  -812  0  2
48A1  0  1  0  -4  -4  0  2 × 2
48A2  0  1  0  -64  -220  0  2
48A3  0  1  0  -24  36  0  4 × 2
48A4  0  1  0  1  0  0  2
48A5  0  1  0  -384  2772  0  4
48A6  0  1  0  16  180  0  8
49A1  1  -1  0  -2  -1  0  2
49A2  1  -1  0  -37  -78  0  2
49A3  1  -1  0  -107  552  0  2
49A4  1  -1  0  -1822  30393  0  2
50A1  1  0  1  -1  -2  0  3
50A2  1  0  1  -126  -552  0  1
50A3  1  0  1  -76  298  0  3
50A4  1  0  1  549  -2202  0  1
50B1  1  1  1  -3  1  0  5
50B2  1  1  1  22  -9  0  5
50B3  1  1  1  -13  -219  0  1
50B4  1  1  1  -3138  -68969  0  1
51A1  0  1  1  1  -1  0  3
51A2  0  1  1  -59  -196  0  1
52A1  0  0  0  1  -10  0  2
52A2  0  0  0  -4  -3  0  2
53A1  1  -1  1  0  0  1  1
54A1  1  -1  0  12  8  0  3
54A2  1  -1  0  -123  -667  0  1
54A3  1  -1  0  -3  3  0  3
54B1  1  -1  1  1  -1  0  3
54B2  1  -1  1  -29  -53  0  1
54B3  1  -1  1  -14  29  0  9
55A1  1  -1  0  -4  3  0  2 × 2
55A2  1  -1  0  -29  -52  0  2
55A3  1  -1  0  -59  190  0  4
55A4  1  -1  0  1  0  0  2
56A1  0  0  0  1  2  0  4
56A2  0  0  0  -19  30  0  2 × 2
56A3  0  0  0  -59  -138  0  2
56A4  0  0  0  -299  1990  0  2
56B1  0  -1  0  0  -4  0  2
56B2  0  -1  0  -40  -84  0  2
57A1  0  -1  1  -2  2  1  1
57B1  1  0  1  -7  5  0  2 × 2
57B2  1  0  1  -2  -1  0  2
57B3  1  0  1  -102  385  0  4
57B4  1  0  1  8  29  0  2
57C1  0  1  1  20  -32  0  5
57C2  0  1  1  -4390  -113432  0  1
58A1  1  -1  0  -1  1  1  1
58B1  1  1  1  5  9  0  5
58B2  1  1  1  -455  -3951  0  1
61a1  1  0  0  -2  1  1  1
62A1  1  -1  1  -1  1  0  4
62A2  1  -1  1  -21  41  0  2 × 2
62A3  1  -1  1  -31  5  0  2
62A4  1  -1  1  -331  2397  0  2
63A1  1  -1  0  9  0  0  2
63A2  1  -1  0  -36  27  0  2 × 2
63A3  1  -1  0  -351  -2430  0  2
63A4  1  -1  0  -441  3672  0  2 × 2
63A5  1  -1  0  -7056  229905  0  4
63A6  1  -1  0  -306  5859  0  2
64A1  0  0  0  -4  0  0  2 × 2
64A2  0  0  0  -44  -112  0  2
64A3  0  0  0  -44  112  0  4
64A4  0  0  0  1  0  0  2

In makeing this table I made use of the following C program.

#include<stdio.h>
main() {
   char s[10][128];
   while(EOF!=scanf("%s%s%s%s%s%s%s%s%s%s",
        s[0],s[1],s[2],s[3],s[4],s[5],s[6],s[7],s[8],s[9]))
      printf("%s%s%s&%s&%s&%s&%s&%s&%s&%s\\\\\n",
        s[0],s[1],s[2],s[3],s[4],s[5],s[6],s[7],s[8],s[9]);
}

********* Begin Unverified *********

10.1.2 Big � ’s

The following curves in the tables of curves of conductor up to 1000 have an “analytic” order of � (E/ℚ) greater than 1.

#[� ] = 4 (67 curves):

66B3 102B5 114C3 120A5 130B3 195A7 210C5 210E5 240D3 258D3 294C3
312C3 330B6 330C6 330D3 336D3 390B5 410B3 435D3 438E2 442A2 442D2
462F3 480B2 480H2 504C3 504G3 510E4 510E7 525B5 558C3 570G3 570I4
570M4 571A1 582B2 582D2 582D4 600D5 630J3 646B2 646E2 646E4 690K4
690K5 714G6 777A4 792E3 798I3 840C3 840G6 840J4 840J6 858E3 858H4
870G3 890H4 910A3 930G4 930O3 960D1 960F3 960G7 960I3 960N1 966I3

990K3

#[� ] = 9 (27 curves):

182B3 300B2 378A3 448C5 448C6 475A3 510G3 510G4 546D3 578A3 578A4
651E3 660D3 660D4 681B1 681B2 681B3 681B4 702P3 714I3 798E5 798E6
903B3 910E3 910J5 910J6 938D3

#[� = 16] (5 curves):

210E7 210E8 582D3 930O5 930O6

#[� = 25] (5 curves):

275B3 570L3 570L4 870I3 870I4

#[� = 49] (2 curves):

546F2 858K2

Status of these results: I have not yet checked that all the supposed generators of each curve really are generators, though they probably are in most cases. If the points(s) I have actually generate a subgroup of index k > 1 in the Mordell-Weil group, then the value of #[� ] should be multiplied by k^{2}. So far I have only checked that none of the points I have is double another, for all conductors except 867, 870, 873, 901, 975.

   JEC 15/4/92 revised 31/12/92

10.2. CURVES OF HIGH RANK 57

10.1.3 Complex multiplication elliptic curves

********* Begin Unverified *********

This is a table of representatives of the complex multiplication elliptic curves over ℚ.

D f  j  curve  model  r t
-3 1  0  27A1 0 0  1  0  -7  0 3
-3 2  54000  36A2 0 0  0  -15  22  0 6
-3 3  −12288000  27A4 0 0  1  -30  63  0 3
-4 1  1728  64A4 0 0  0  1  0  0 2
-4 2  287496  32A4 0 0  0  -11  14  4 4
-7 1  −3375  49A1 1 -1  0  -2  -1  0 2
-7 2  16581375  2^{4}7^{2}H2 0 0  0  -595  5586  1 2
-8 1  8000  256A1 0 1  -3  1  4  1 2
-11 1  −32768  11^{2}B 0 -1  1  -7  10  1 1
-19 1  −884736  19^{2}A1 0 -1  1  -7  10  1 1
-43 1  −884736000  43^{2}A1 0 0  1  -860  9707  1 1
-67 1  −147197952000  67^{2}A1 0 0  1  -7370  243528  1 1
-163 1 −262537412640768000 163^{2} 0 0  1  -2174420 1234136692 ≥ 1 1

Question 10.1.1. What is the rank of the last CM curve on the above list? The conductor is 163^{2} = 26569 which is out of the range of cremona’s tables. I ran mwrank for three days on bosco.berkeley.edu and it did not terminate. It did find the point (850, 68). This point has a height (according to hell in pari) of 1.14564... so it has infinite order. Thus the rank is ≥ 1. Does this point generate the full group of rational points?

This table contains all CM curves of conductor ≤ 5077.

j  curves
0  27A1 27A3 36A1 36A3 108A1 108A2 144A1 144A3 576A1 576A3 576E1 576E3 675A1 675A2 675C1 675C2 675E1 675E2

900A1 900A2 900B1 900B3 900C1 900C2 972A1 972A2 972B1 972B2 972C1 972C2 972D1 972D2 1089A1 1089A2 1089B1 1089B2 1323A1 1323A2 1323B1 1323B3 1323M1 1323M2 1521A1 1521A2 1521B1 1521B2 1728A1 1728A2 1728F1 1728F2 1728M1 1728M2 1728V1 1728V3 1764A1 1764A2 1764B1 1764B3 1764C1 1764C2 225A1 225A2 225B1 225B2 243A1 243A2 243B1 243B2 2700E1 2700E2 2700G1 2700G2 2700H1 2700H2 2700L1 2700L2 2700P1 2700P2 2700T1 2700T2 2700U1 2700U2 3267A1 3267A2 3267B1 3267B2 3267D1 3267D3 3600B1 3600B3 3600C1 3600C2 3600D1 3600D2 3600E1 3600E2 3600Y1 3600Y2 3888K1 3888K2 3888L1 3888L2 3888M1 3888M2 3888P1 3888P2 3888S1 3888S2 3888T1 3888T2 432A1 432A3 432B1 432B2 4356A1 4356A2 4356B1 4356B2 4356C1 4356C3 441A1 441A2 441B1 441B2 4563A1 4563A2 4563B1 4563B2 4563G1 4563G2 4761A1 4761A2 4761B1 4761B2

54000  36A2 36A4 144A2 144A4 576A2 576A4 576E2 576E4 900B2 900B4 1764B2 1764B4 3600BB2 3600BB4 4356C2 4356C4
-12288000  27A2 27A4 432A2 432A4 675A3 675A4 1323B2 1323B4 1728A3 1728A4 1728V2 1728V4 3267D2 3267D4 4563G3 4563G4
1728  32A1 32A2 64A1 64A4 256B1 256B2 256C1 256C2 288A1 288A2 288D1 288D4 288E1 288E2 576F1 576F2 576G1

576G2 576H1 576H2 800A1 800A4 800D1 800D2 800H1 800H2 1568A1 1568A2 1568B1 1568B2 1568G1 1568G4 1600O1 1600T1 1600T2 1600U1 1600U2 2304A1 2304A2 2304B1 2304B2 2304H1 2304H2 2304I1 2304I2 2304J1 2304J2 2304P1 2304P2 3136T1 3136T2 3136U1 3136U2 3136V1 3136V2 3872A1 3872A2 3872B1 3872B4 3872H1 3872H2

287496  32A3 32A4 64A2 64A3 288D2 288D3 576H3 576H4 800A2 800A3 1568G2 1568G3 1600O3 1600O4 3136T3 3136T4

3872B2 3872B3

-3375  49A1 49A3 441D1 441D3 784H1 784H3 1225B1 1225B3 3136D1 3136D3 3136R1 3136R3
16581375  49A2 49A4 441D2 441D4 784H2 784H4 1225B2 1225B4 3136D2 3136D4 3136R2 3136R4
8000  256A1 256A2 256D1 256D2 2304C1 2304C2 2304K1 2304K2
-32768  121B1 121B2 1089E1 1089E2 1936F1 1936F2 3025A1 3025A2
-884736  361A1 361A2 3249A1 3249A2
-884736000  1849A1 1849A2
-147197952000  4489A

********* End Unverified *********

10.2 Curves of high rank

10.3 Weierstrass Equations

********* Begin Unverified *********

10.3.1 Finding weierstrass form of an elliptic curve

Plane Cubic

The Weierstrass form of an elliptic curve is

y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6}.

The Weierstrass form is homogeneous of degree 6 if y is given weight 3, x is given weight 2, and a_{i} is given weight i.

The genus one curve defined by a nonsingular Weierstrass equation has a rational point at infinity. Because the Selmer curve defined by 3x^{3} + 4y^{3} + 5 = 0 has no rational points we see that not every cubic equation can be put in Weierstrass form.

The cubic

ay^{2} = bx^{3} + cx^{2} + dx + e

can be transformed into the Weierstrass form

y_{1}^{2} = x_{1}^{3} + \frac{c}{a}x_{1}^{2} + \frac{db}{a^{2}} x_{1} + \frac{eb^{2}}{a^{3}}

by the transformation

x ↦ \frac{a}{b}x_{1}, y ↦ \frac{a}{b}y_{1}.

The Fermat cubic X^{3} + Y^{3} = Z^{3}

Proposition 10.3.1. The elliptic curve defined by x^{3} + y^{3} = 1 has Weierstrass equation y^{2} = x^{3} − \frac{27}{4}.

Proof. Make the substitution x ↦ x − y to obtain

1 = (x − y)^{3} + y^{3} = x^{3} − 3x^{2}y + 3xy^{2}.

Dividing through by x^{3} then gives

\frac{1}{x^{3}} = 1 − 3\frac{y}{x} + 3(\frac{y}{x})^{2}

The substitution x ↦ 1/x, y ↦ y/x transforms this into 3y^{2} − 3y = x^{3} − 1. Now multiply both sides by 27 to obtain

(9y)^{2} − 9(9y) = (3x)^{3} − 27

which, upon performing the substitution y ↦ 9y, x ↦ 3x becomes y^{2} − 9y = x^{3} − 27. Now complete the square by adding \frac{81}{4} to both sides to obtain

(y − \frac{9}{2})^{2} = x^{3} − 27 + \frac{81}{4} = x^{3} − \frac{27}{4}.

The substitution y − \frac{9}{2} ↦ y completes the proof.

In fact the curve x^{3} + y^{3} = z^{3} is 27A1 in Cremona’s tables and has Weierstrass minimal model y^{2} + y = x^{3} − 7. In this Weierstrass form, E(ℚ) = {(3, 4), (3, −5), ∞}.

Intersection of Two Quadrics in ℙ^{3}

[Refer to y^{2} = −(x^{4} + 1) example. Outline general strategy of “effective Riemann-Roch”.]

10.3.2 Invariants attached to weierstrass models

The discriminant of the above Weierstrass equation is

Δ = −b_{2}^{2}b_{8} − 8b_{4}^{3} − 27b_{6}^{2} + 9b_{2}b_{4}b_{6}

where

b_{2} = a_{1}^{2} + 4a_{2}, b_{4} = a_{1}a_{3} + 2a_{4}, b_{6} = a_{3}^{2} + 4a_{6}
b_{8} = a_{1}^{2}a_{6} + 4a_{2}a_{6} − a_{1}a_{3}a_{4} + a_{2}a_{3}^{2} − a_{4}^{2}

The j-invariant is

j = (b_{2}^{2} − 24b_{4})^{3}/Δ.

For the Weierstrass equation y^{2} = x^{3} + ax + b we have

Δ = −16(4a^{3} + 27b^{2}), j = 12^{3} \frac{4a^{3}}{4a^{3} + 27b^{2}} .

The j-invariant of the curve defined by y^{2} = 4x^{3} + ax + b is \frac{1728a^{3}}{a^{3}+27b^{2}} . To see this divide by 4 and consider (\frac{y}{2} )^{2} = x^{3} + \frac{a}{4}x + \frac{b}{4}.

A Weierstrass equation for an elliptic curve is unique up to a coordinate transformation of the form

x ↦ u^{2}x′ + r, y ↦ u^{3}y′ + su^{2}x′ + t, u, r, s, t ∈ k, u ≠ 0

If Δ′ is the discriminant of the transformed equation then Δ = u^{12}Δ′. Thus the discriminants of two different models for the same curve differ by a twelfth power.

A minimal Weierstrass equation is one for which a_{i} ∈ ℤ and Δ is minimal. It is unique up to a coordinate transformation of the above form with r, s, t, u ∈ ℤ, u ∈ ℤ^{×} = {±1}. The minimal discriminant need not be the conductor N_{E} of the corresponding elliptic curve E. The two quantities are related because

N_{E} = ∏_{p} p^{ord_{p}(Δ)+1−m_{p}}

where m_{p} is the number of connected components of the Néron model of E at p, and the product is over all finite primes p of ℚ. A Weierstrass model of E is never a Néron model of E. [If it were then E would have nonsingular reduction everywhere which can not happen.] It can be the case that N_{E} = |Δ|, this occurs for 11A3 (y^{2} + y = x^{3} − x^{2}) which has discriminant −11. But for 11A1 the minimal discriminant is −11^{5}.

Example 10.3.2. The Weierstrass equation

y^{2} = x^{3} + \frac{1}{4}x

has discriminant

Δ = −16(4(\frac{1}{4})^{3}) = −1.

Note that this is not a minimal Weierstrass equation and that the minimal Weierstrass equation corresponding to the elliptic curve defined by the above equation has discriminant −2^{12}. This example shows that if the Weierstrass equation is not minimal then the set of primes dividing the discriminant may be completely disjoint from the set of primes of bad reduction.

60 CHAPTER 10. ELLIPTIC CURVES

10.4 The Birch and Swinnerton-Dyer Conjecture

10.4.1 The L-series attached to an elliptic curve

Definition of L(E, s)

********* Begin Unverified *********

Let E be an elliptic curve over ℚ. For each prime p of ℚ define

L_{p}(T) = \begin{cases}1 − a_{p}T + pT^{2} & a_{p} = p + 1 − N_{p}, p good \\ 1 − T & if E has split multiplicative reduction \\ 1 + T & if E has non-split multiplicative reduction \\ 1 & if E has additive reduction \end{cases}

The L-series of E is now defined to be

L(E, s) = ∏_{p} \frac{1}{L_{p}(p^{−s})}.

Basic Facts about L(E, s) Theorem 10.4.1. L(E, s) converges for ℜ(s) > \frac{3}{2}. Theorem 10.4.2. The elliptic curves E and E′ are isogenous iff L(E, s) = L(E′, s).

Numerically approximating L(E, s) + what I know about using pari to do this + modular symbols algorithm ?

10.4.2 The Birch and Swinnerton-Dyer Conjecture

Fix an elliptic curve E over ℚ. Let • R be the regulator of E,

• Ω = ∫_{E(ℝ)} |ω| where ω is the invariant Néron differential,

• c_{p} the local Tamagawa number of E at p,

• N the conductor of E,

• � the Tate-Shafarevich group of E,

• r the rank of E(ℚ), and

• L(E, s) the L-series of E. Birch and Swinnerton-Dyer conjecture that these quantities are related to the residue at s = 1 of L(E, s). Conjecture 10.4.3 (Birch and Swinnerton-Dyer).

lim_{s→1} \frac{L(E, s)}{(s − 1)^{r}} = \frac{R · Ω · ∏_{p|N} c_{p} · [� ]}{[E(ℚ)_{tor}]^{2}}

10.4.3 The Tamagawa numbers

********* Begin Unverified *********

Let E be an elliptic curve over ℚ and define

c_{p} = (E(ℚ_{p}) : E^{0}(ℚ_{p})).

Thus c_{p} is the number of connected components of the special fibre of the Néron model at p.

The local Tamagawa number c_{p} can be computed using PARI, it is localred(e,p)[4]. The curve y^{2} = x^{3} − x has conductor 32, c_{2} = 2, and c_{p} = 1 (p ≠ 2).

? e=initell([0,0,0,-1,0])
? localred(e,2)
%1 = [5, 3, [1, 1, 1, 0], 2]
? localred(e,3)
%2 = [0, 1, [1, 0, 0, 0], 1]

The global Tamagawa number c is the product of the local Tamagawa numbers. PARI computes

it as globalred(e,p)[3].
? globalred(e)
%3 = [32, [1, 0, 0, 0], 2]

********* End Unverified *********

10.4.4 The Néron-Tate canonical height

********* Begin Unverified *********

Definition

Let E be an elliptic curve over ℚ embedded in ℙ^{2} by choice of a Weierstrass equation

y^{2} = x^{3} + ax + b, a, b ∈ ℚ.

For P ∈ E(ℚ) define

h(P) = \begin{cases}0 & if P = ∞ \\ log(max(c, d)) & if P = (x, y), |x| = \frac{c}{d} \end{cases}

where c and d are relatively prime positive integers. The Néron-Tate canonical height is

ĥ(P) = lim_{n→∞} \frac{h(2^{n}P)}{2 · 4^{n}}.

Computation

The gp function hell(e,P) can be used to approximate ĥ(P).

? h(p)=log(max(abs(numer(p[1])),abs(denom(p[1]))))
? hhat(e,p,n)=h(powell(e,p,2^n))/(2*4^n)
? e=initell([0,0,1,-1,0]);
? hhat(e,[0,-1],5)
%1 = 0.02555039173998061935821379001
? hell(e,[0,-1])
%2 = 0.02555570411998442011794304989

Standard Facts

Theorem 10.4.4. There are only finitely many points P ∈ E(ℚ) with height less than a fixed constant C.

Proposition 10.4.5. A point P ∈ E(ℚ) has height zero iff P ∈ E(ℚ)_{tor}.

The regulator

Define

B(P, Q) = ĥ(P + Q) − ĥ(P) − ĥ(Q).

Then B is a symmetric bilinear form. Suppose E(ℚ) has rank r and let P_{1}, …, P_{r} be a basis, modulo torsion. Define the regulator of E to be the determinent

R = det(B(P_{i}, P_{j})).

[If P_{1}, …, P_{s} ∈ E(ℚ) then the rank of the matrix (B(P_{i}, P_{j})) gives the rank of the set of points.]

[???] Is there a way to compute the regulator without computing a basis for E(ℚ)? Milne asserts on page 104 of his notes on elliptic curves that every quantity (except � ) appearing in the BSD conjecture is effectively computable. I don’t think E(ℚ) is known to be effectively computable so either Milne’s statement is false or there is a better way to compute the regulator.

The PARI functions bilhell and mathell can be used to work with the height pairing. Let E be the curved defined by y^{2} + xy = x^{3} + 1. From Cremona’s table we see that E is 433A1 and that E(ℚ) ≅ ℤ × ℤ. One sees that (−1, 1) and (−1, 0) are two rational points on E. Are they independent?

? e=initell([1,0,0,0,1]);
? det(mathell(e,[[-1,1],[-1,0]]))
%1 = 6.310887241736054420000000000 E-30

Thus the two points are not independent. Another rational point on E is (1, 1).

? det(mathell(e,[[1,1],[-1,0]]))
%2 = 0.2246941634181667416120711342

Thus (1, 1) and (−1, 0) are independent. [???] How can I tell whether or not (1, 1), (−1, 0) form a basis for E(ℚ)?

********* End Unverified *********

10.4.5 The real period of an elliptic curve

Let E be an elliptic curve with minimal Weierstrass equation

y^{2} + a_{1}xy + a_{3}y = x^{3} + a_{2}x^{2} + a_{4}x + a_{6}.

The invariant Néron differential on E is

ω = \frac{dx}{2y + a_{1}x + a_{3}} .

The real period of E is

ω_{1} = ∫_{E(ℝ)^{0}} |ω|

where E(ℝ)^{0} is the connected component of the identity of E(ℝ). Since the group law is a diffeomorphism,

Ω = ∫_{E(ℝ)} |ω| = \begin{cases}ω_{1} & if E(ℝ) has one component \\ 2ω_{1} & if E(ℝ) has two components \end{cases}

PARI can be used to approximate ω_{1}. If e is the elliptic curve data returned by initell, then e[15] is approximately equal to ω_{1}. It is also possible to directly compute ω_{1} using numerical integration. For example, consider the curve defined by y^{2} = x^{3} − x. Then the integral

∫_{E(ℝ)^{0}} |ω| = 2 ∫_{1}^{∞} \frac{dx}{\sqrt{x^{3} − x}}

can be numerically approximated with PARI.

? \precision=5
? intinf(x=1,1e4000,1/sqrt(x^3-x))
%1 = 2.5625
? initell([0,0,0,-1,0])[15]
%2 = 2.6220

The number ω_{1} is not a \overline{ℚ}-isomorphism invariant. For example, the curve E′ : y^{2} = x^{3} − 4x is isomorphic over \overline{ℚ} to E : y^{2} = x^{3} − x (they both have j-invariant 1728), but ω_{1}(E′) ≈ 1.854 whereas ω_{1}(E) ≈ 2.622.

ω_{1} is called the real period because there is a complex number ω_{2} such that ℤω_{1} + ℤω_{2} is the complex lattice associated to E.

10.5 Elliptic Curves over Finite Fields

10.5.1 Hasse Bound

Theorem 10.5.1 (Hasse). Let E be an elliptic curve over the finite field 𝔽_{q} having q elements. Then

|[E(𝔽_{q})] − (q + 1)| ≤ 2\sqrt{q}.

10.5.2 What values in the Hasse interval are attained?

********* Begin Unverified *********

The following pari program enumerates the Weierstrass equations which define elliptic curves over 𝔽_{2} along with the size of their group of rational points.

? discell(a1,a2,a3,a4,a6,b2,b4,b6,b8)=b2=a1^2+4*a2;b4=a1*a3+2*a4;b6=a3^2+4*a6;\
  b8=a1^2*a6+4*a2*a6-a1*a3*a4+a2*a3^2-a4^2;-b2^2*b8-8*b4^3-27*b6^2+9*b2*b4*b6
? ordf2(e,c)=c=1;for(x=0,1,for(y=0,1,if(isoncurve(e,[mod(x,2),mod(y,2)]),c=c+1,)));c
? for(a1=0,1,for(a2=0,1,for(a3=0,1,for(a4=0,1,\
  for(a6=0,1,if(mod(discell(a1,a2,a3,a4,a6),2)==0,,\
  print(a1,a2,a3,a4,a6,": N=",ordf2(initell([a1,a2,a3,a4,a6])))))))))

The output is (reformatted to save space):

00100: N=3, 00101: N=3, 00110: N=5, 00111: N=1, 01100: N=5, 01101: N=1,
01110: N=3, 01111: N=3, 10001: N=4, 10010: N=4, 10101: N=2, 10111: N=2,
11001: N=2, 11010: N=2, 11100: N=4, 11110: N=4

The Hasse theorem asserts that if E is an elliptic curve over 𝔽_{2} then |E(𝔽_{2})−3| ≤ 2\sqrt{2} ≈ 2.828, i.e., the possible values for E(𝔽_{2}) are 1, 2, 3, 4, or 5. For q = 2, we see that all values occur.

Using the following program we can investigate the situation for more general p.

? ordfp(e,p,c)=c=1;for(x=0,p-1,for(y=0,p-1,\
  if(isoncurve(e,[mod(x,p),mod(y,p)]),c=c+1,)));c
? allcurves(p)=for(a1=0,p-1,for(a2=0,p-1,for(a3=0,p-1,for(a4=0,p-1,\
  for(a6=0,p-1,if(mod(discell(a1,a2,a3,a4,a6),p)==0,,\
  print(a1,a2,a3,a4,a6,": N=",ordfp(initell([a1,a2,a3,a4,a6]),p))))))))

? allcurvesN(p)=for(a1=0,p-1,for(a2=0,p-1,for(a3=0,p-1,for(a4=0,p-1,\
  for(a6=0,p-1,if(mod(discell(a1,a2,a3,a4,a6),p)==0,,\
  print(ordfp(initell([a1,a2,a3,a4,a6]),p))))))))

I tried allcurvesN(p) for the primes p = 2, 3, 5, 7 and in each case all values in the Hasse interval were quickly attained. I don’t know what happens in the general situation.

Computation

11.1 Pari

********* Begin Unverified *********

11.1.1 Compiling Pari under Linux

I successfully compiled pari version 1.39.03 under slackware linux by performing the following steps.

1. Create the makefile Makefile by typeing Makemakefile i386.

2. Modify Makefile by replacing version386.c on lines 164 and 165 by versionport.c. [This might be a non-optimal fix.]

3. Type make in the src directory to build everything. This takes about one hour.

The command make install successfully installed pari and the documentation.

********* End Unverified *********

11.1.2 Library Mode

********* Begin Unverified *********

Makefile

Here is a Makefile which allows my system to compile the C program shell.c and the C++

program shellcpp.cpp.
CC = gcc
CPP = g++
CFLAGS = -O2 -ansi -I/usr/local/include/pari -DLONG_IS_32BIT
CPPFLAGS = -O2 -I/usr/local/include/pari -DLONG_IS_32BIT
all:    shell shellcpp
shell: shell.c
$(CC) $(CFLAGS) -o shell shell.c -lpari -lm
shellcpp:  shellcpp.cpp
$(CPP) $(CPPFLAGS) -o shellcpp shellcpp.cpp -lpari -lm

Sample C Program

/* shell.c -- a very simple pari shell. */
#include <genpari.h>
main() {
   GEN x; char s[512];
   init(4000000,100000);   /* Initialize the pari stack. */
   printf("Simple pari shell.\n> ");
   while(EOF!=scanf("%s",s)) {
      x=lisseq(s);   /* input and parse a list of expressions */
      sor(x,'f',-1,0);   /*  output the result */
      printf("\n> ");
   } }

Sample C++ Program

// shellcpp.cpp -- a very simple pari shell in C++.
#include <iostream.h>
extern "C" {
#include <genpari.h>
}
main() {
   init(4000000,100000);   // Initialize the pari stack.
   cout << "Simple pari shell." << endl << "> ";
   GEN x; char s[512];
   while(EOF!=scanf("%s",s)) {
      x=lisseq(s);          // input and parse a list of expressions
      sor(x,'f',-1,0);      // output the result
      cout << endl << "> ";
} }

********* End Unverified *********

11.1.3 gp Shell

11.1.4 Numerical Integration

********* Begin Unverified *********

PARI implements basic numerical integration.

• intnum(x=a,b,f(x)): simple numerical evaluation of ∫_{a}^{b} f(x). Use when a and b are small and f(x) is smooth and well-defined on [a, b].

• intinf(x=a,b,f(x)): numerical integration optimized for when a or b is infinite. To PARI infinite means 1e4000. a and b must have the same sign.

• intgen(x=a,b,f(x)): general numerical evaluation of ∫_{a}^{b} f(x). The limits a and b can have different signs and can be large.

The integrand f(x) can be complex valued.

? \precision=3
? intgen(t=0,1,(1+i)*t)
%1 = 0.500 + 0.500*i

If f(x) is not continuous and the precision is not small then the PARI stack may overflow.

? \precision=3
? f(x)=if(x<0,-1,1)
? intgen(x=-1,1,f(x))
%1 = 0.0000740
? \precision=28
? intgen(x=-1,1,f(x))
  ***   the PARI stack overflows !!!

Here are some other examples.

? \precision=3
? intnum(t=1,10,1/t^3)
%1 = 0.495
? intinf(t=1,1e4000,1/t^3)
%2 = 0.500
? intgen(t=-1,1e4000,1/(t^2+1))
%3 = 2.35
? intinf(t=-1,1e4000,1/(t^2+1))   // WRONG!
%4 = -0.785

Note that the last usage of intinf is erroneous because the limits of integration are not of the same sign.

********* End Unverified *********

11.2 LiDIA

********* Begin Unverified *********

LiDIA is a C++ library for number theory. It implements

11.2.1 Examples

nextprime.C

// nextprime.C -- print out the prime following a certain prime.
// [see page 30 of the LiDIA man.dvi documentation]
#include<LiDIA/bigint.h>
main() {
   bigint p, next;
   // let p be the prime 2^61 - 1.
   string_to_bigint("2305843009213693951",p);
   // let next=the next prime after p.
   next=next_prime(p);
   cout<<"The next prime after 2^61-1 is "<<next<<"."<<endl;
}

********* End Unverified *********

11.3 Cremona’s programs: mwrank, etc.

J. Cremona has constructed programs for finding the group of rational points on an elliptic curve. They can be obtained at

                    ftp://euclid.ex.ac.uk/pub/cremona/progs/

To get an idea of how quickly various computers I use can run mwrank I asked it to compute the Mordell-Weil group of the curve y^{2} + y = x^{3} − 199x + 1092. Here are the running times.

Computer Architecture Time (seconds)
june’s  P233MMX  9
number  P200  14
whale  P166  16
ferrari  pentium  20
sizzlean  sparc 4  54
jaffna  sparc  88
bosco  sparc  91

11.3.1 Bug Report

I sent the falling “bug report” to pari@math.u-bordeaux.fr on July 20, 1997.

Dear PARI developers,
I was constructing a table of class groups of quadratic imaginary extensions
when I received a message asking me to please report a problem.
When I issued the command
        buchgenfu(x^2+46957)
gp worked for a long time then finally printed:
 ***   sorry, buchxxx is not able to compute this field PLEASE REPORT!!!.
Note that 46957 is prime. The next prime after 46957 is 46993, so
I tried buchgenfu(x^2+46993) and it worked fine.
I am using an intel pentium 200 but I duplicated the problem on
a sparc.  A complete transcript follows my signature.
I hope I have been of some help,
William Stein (was@math.berkeley.edu)
-------------------------------------------------
# This is on my intel P200 running Linux 2.0.30.  (I compiled this
  build of pari myself.)
/usr/local/bin/gp -s 10000000 -b 30000 -p 500000
            GP/PARI CALCULATOR Version 1.39.03
                 (portable 32-bit version)
Copyright 1989-1995 by C. Batut, D. Bernardi, H. Cohen and M. Olivier
Type ? for help
\precision      = 28
\serieslength   = 16
\format         = g0.28
\prompt         = ?
stacksize = 10000000, prime limit = 500000, buffersize = 30000
? buchgenfu(x^2+46957)
  ***   sorry, buchxxx is not able to compute this field PLEASE REPORT!!!.

? buchgenfu(x^2+46993)
%1 =
[x^2 + 46993] [[0, 1]] [[-187972, 1]] [[1, x]] [[56, [56], [[43, 36; 0, 1]]]]
[1][1.028570398514380248352478863] [[2, -1]] [[]] [-1073744772] (I removed newlines)

# This is on one of the sparcs in the math department.  (I downloaded
# this build of pari from a web site.)
beirut 25->gp -s  10000000
            GP/PARI CALCULATOR Version 1.39
                     (Sparcv7 version)
Copyright 1989-1994 by C. Batut, D. Bernardi, H. Cohen and M. Olivier
Type ? for help
\precision      = 28
\serieslength   = 16
\format         = g0.28
\prompt         = ?
stacksize = 10000000, prime limit = 500000, buffersize = 30000
? buchgenfu(x^2+46957)
  ***   sorry, buchxxx is not able to compute this field PLEASE REPORT!!!.
-------------------------------------------------

11.4 GAP

GAP – Groups Algorithms Programming

********* Begin Unverified *********

11.4.1 Examples

Lattice of subgroups

Here is how to use GAP to find the lattice of subgroups of the group S_{4}.

gap> s4:=Group((1,2,3,4),(1,2));;
gap> lat:=Lattice(s4);
gap> SetPrintLevel(lat,2);
gap> lat;
   .. a list of the subgroups
gap> SetPrintLevel(lat,3);
gap> PrintClassSubgroupLattice(lat,9);
   .. a detailed description of all subgroups of order 8.
   .. the "9" means the ninth class.

Unix Notes

12.1 Scripts

********* Begin Unverified *********

12.1.1 pack and unpack.

It is useful to have two shell scripts pack and unpack defined as follows. Running pack creates two files book.tgz and bookMD.tgz (where M and D denote the month and day, respectively) which contains all .tex files in the directory book. It then copies book.tgz to the A: floppy drive

and copies bookMD.tgz to A:\dated. Running unpack first makes a backup .pack backup.tgz

of your current files, then copies the file book.tgz from the A: disk, and finally decompresses

book.tgz.

A shell script implementation of pack is:

cd ~/book
fname=`date +"book%m%d%.tgz"`
tar -czvf $fname *.tex pack unpack
mcopy -on $fname a:\dated
mcopy -on $fname a:book.tgz
   unpack is:
cd ~/book
mcopy -on a:\book.tgz .
tar -czvf .pack_backup.tgz *.tex pack unpack
tar -xzvf book.tgz

********* End Unverified *********

********* Begin Unverified *********

12.1.2 Check spelling

The following shell script checks the spelling of all of its arguments.

       echo "$*" > /tmp/a ; ispell /tmp/a

********* End Unverified *********

12.1.3 /etc/profile

I made the following changes and additions to the standard slackware /etc/profile. The line

 PS1='\h:\w\$ '

is changed to

 '`whoami`:\w\$ '

because it is more important to know the current user than my machine name. My list of aliases is:

alias lpps='gs -dNOPAUSE -sDEVICE=ljet4 -sOutputFile=/dev/lp1'
alias lpdvi='dvilj4 -e/dev/lp1'
alias sizzlean='telnet sizzlean.berkeley.edu'
alias f='telnet ferrari.autobahn.org'
alias bmw='telnet bmw.autobahn.org'
alias x='startx'
alias d='disconnect'     # disconnect is a program I wrote.
alias p='ps -a'
alias a='autobahn'

12.1.4 Configuring X Windows

Wall Paper

The following command makes pic.jpg the wallpaper.

xv -root -viewonly pic.jgg

12.1.5 Trash can

12.1.6 Useful Aliases

12.2 Networking

********* Begin Unverified *********

12.2.1 Windowing programs over the net.

If I am sitting in the math department running X-windows on sphynx.berkeley.edu and I want to window back a program from my home computer dialup41-berkeley.autobahn.org (say xpaint, or emacs), then I do the following. First, on sphynx I execute the command xhost +dialup41-berkeley.autobahn.org. Then, I open up a telnet connection to my home computer and type xpaint. In a minute or two, xpaint appears on my desktop. It is very slow, but useable.

12.3 TEX

12.3.1 Characters

To typeset �

   \font\cyr=wncyr10   % will \magstep 1 work for 10pt?
   \newcommand{\Sha}{\hbox{\cyr X}}

To typeset semidirect products ⋉, ⋊

The program

\documentclass{article} \usepackage{amsfonts}
\begin{document} $$H\Bbb n N$$ $$N\Bbb o H$$ \end{document}

produces the output

H⋉N
N⋊H

Radicals

Radicals a typeset poorly in L^{A}TEX but AMS-L^{A}TEX offers the following which looks better

\sqrt[\leftroot{2}\uproot{4}\beta]{k}

Boxed Formulae

The command boxed puts a box around its arguments.

********* End Unverified *********

********* Begin Unverified *********

12.4 Conditionals

12.4.1 Conditionals in L^{A}TEX

% this is an example of the use of conditionals in LaTeX.
\documentclass{article}
\usepackage{ifthen}
\begin{document}
% test if two strings are equal.
\newcommand{\username}{William}
You \ifthenelse{\equal{\username}{William}}{are}{are not} William.
% use boolean variables in tests.
\newboolean{bool}
\setboolean{bool}{true}
{\tt bool} is \ifthenelse{\boolean{bool}}{true}{false}.
\setboolean{bool}{false}
{\tt bool} is \ifthenelse{\boolean{bool}}{true}{false}.
% combining conditions: use \and \or \not \( \)
\ifthenelse{\(1=1\) \and \( \not \boolean{bool} \)}{yes}{no}.
% make a ``comment out'' macro.
\newboolean{printcomments}
\setboolean{printcomments}{false}
\newcommand{\comment}[1]{\ifthenelse{\boolean{printcomments}}{#1}{}}
\comment{I don't believe it!}
\setboolean{printcomments}{true}

\comment{The {\em real} subtlety is that...}
\end{document}

12.4.2 Conditionals in TEX

% this is an example of the use of conditionals in TeX.
hello
\ifnum 1<2
   order
\else
   disorder
\fi
% to facilitate \if... constructions, plain TeX has a \newif macro, such
% that after you say `\newif\ifabc' three control sequences will be defined:
% \ifabc (for testing the switch), \abctrue (for making the switch true),
% and \abcfalse (for making it false).
\newif\ifabc
\def\status{abc is \ifabc true \else false\fi}
\abctrue \status
\abcfalse \status
\bye

********* End Unverified *********

12.5 Emacs

At the time of this writing I am using GNU Emacs, version 19.34. Here is my .emacs file.

; always use LaTeX and never TeX mode.
(setq auto-mode-alist (cons '("\.tex$" . latex-mode) auto-mode-alist))
(setq tex-dvi-view-command "xdvi -s 3 -geometry 860x965+408+2 ")
(define-key esc-map "f" 'font-lock-mode)   ; colorful fonts
(define-key esc-map "l" 'auto-fill-mode)   ; auto-CR at end of line
(define-key esc-map "g" 'goto-line)
(define-key esc-map "r" 'query-replace)
; PARI
(autoload 'gp "pari" " pari-gp" t)
(autoload 'gpman "pari" " pari-gp manual" t)
; Macaulay 2
(setq auto-mode-alist (append auto-mode-alist '(("\\.m2$" . M2-mode))))
(autoload 'M2-mode "M2-mode.el" "Macaulay 2 editing mode" t)
(global-set-key "\^Cm" 'M2) (global-set-key [ f12 ] 'M2)
(autoload 'M2 "M2.el" "Run Macaulay 2 in a buffer." t)
(setq load-path (cons "/usr/local/Macaulay2/emacs" load-path))
(setq transient-mark-mode t)

Euler Systems

13.1 Rubin, The work of Kolyvagin on the arithmetic of elliptic curves

********* Begin Unverified *********

Modulo details, Rubin gives a simplified version of Kolyvagin’s proof of finiteness of E(ℚ) and � (E/ℚ) for modular elliptic curves having analytic rank 0. The exposition is very terse and many details are omitted. In fact certain key details remain hidden from me still.

********* End Unverified *********

********* Begin Unverified *********

13.1.1 Introduction

This may be considered as an appendix to Rubin, The work of Kolyvagin on the arithmetic of elliptic curves, Springer Lecture Notes in Math, 1399. This arose out of a seminar organized by Keving Buzzard at Berkeley during the Spring of 1997.

To simplify notation and fix ideas we prove only the following weaker result.

Theorem 13.1.1 (Kolyvagin). Let E/ℚ be a modular elliptic curve such that L(E, 1) ≠ 0. Then E(ℚ) is finite and � (E/ℚ)[p] is finite for all sufficiently large p.

We always assume that p > 2 is a prime and that K ≠ ℚ(\sqrt{−1}), ℚ(\sqrt{−3}).

13.1.2 Group Cohomology

Let K be a field and let A be a Gal(K̄/K)-module. Let L be a finite Galois extension of K. The restriction-inflation sequence is:

0 → H^{1}(L/K, A) ⟶^{inf} H^{1}(K, A) ⟶^{res} H^{1}(L, A).

Also Gal(L/K) acts on H^{1}(K̄/L, A) as follows. Fix φ ∈ Gal(L/K) and an element of H^{1}(K̄/L, A) represented by a cocycle f. Then

φ ∗ f(σ) = \overline{φ}f((\overline{φ})^{−1}σ\overline{φ})

where \overline{φ} is some lift of φ to Gal(K̄/K). Although this action on cocycles depends on the choice of a lift of ϕ, the induced action on cohomology does not.

Proposition 13.1.2. Let G be a group and let A be a G-module. Suppose n is a positive integer such that multiplication by n induces an isomorphism n : A ⟶^{≅} A. Then H^{i}(G, A)_{n} = 0 for all i ≥ 0.

13.1. RUBIN, THE WORK OF KOLYVAGIN ON THE ARITHMETIC OF ELLIPTIC CURVES75

Proof. Consider the exact sequence

0 → 0 → A ⟶^{n} A → 0.

Taking cohomology yields, for each i, an exact sequence

0 = H^{i}(G, 0) → H^{i}(G, A) ⟶^{n} H^{i}(G, A) → H^{i+1}(G, 0) = 0

so H^{i}(G, A)_{n} = 0. Caution. If multiplication by n is either injective or surjective, but not both, then H^{i}(G, A)_{n} may not vanish.

13.1.3 Application of Tate Local Duality

Proposition 13.1.3. Suppose ℓ is a prime such that E(ℚ_{ℓ})_{p} ≅ ℤ/pℤ. If there exists c ∈ H^{1}(ℚ, E)_{p} such that

1. for all v ≠ ℓ, res_{v}(c) = 0,

2. res_{l}(c) ≠ 0 Then res_{l}(Sel_{p}(E/ℚ)) = 0.

13.1.4 The Euler System of Heegner points 13.1.5 Cohomology Classes Arising from Heegner points

Lemma 13.1.4. Suppose ℓ is a prime not dividing pD_{K}N and Frob_{ℓ}(K(E_{p})/ℚ) = [τ]. Then if Ẽ denotes the reduction of E modulo ℓ and a_{ℓ} = ℓ + 1 − #[Ẽ(𝔽_{ℓ})],

1. p|a_{ℓ} and p|ℓ + 1,

2. ℓ remains prime in K,

3. E(ℚ_{ℓ})_{p} ≅ Ẽ(𝔽_{ℓ})_{p} ≅ ℤ/pℤ, and

4. (E(K_{ℓ})_{p})^{−} ≅ (Ẽ(𝔽_{l^{2}})_{p})^{−} ≅ ℤ/pℤ.

[DEFINE THE HEEGNER POINTS HERE.]] Proposition 13.1.5. For ℓ ∤ N, the Heegner point y_{l} satisfies the following. ES1. Tr_{K[l]/H}(y_{ℓ}) = a_{ℓ}y_{H}. ES2. For any prime λ of K[ℓ] above ℓ,

ỹ_{l} = Frob(t̂) ∈ Ẽ(𝔽_{ℓ^{2}}),

where Ẽ denotes reduction modulo λ and Frob is the ℓ-th power map on the coordinates. Suppose ℓ is a prime not dividing pD_{K}N and Frob_{ℓ}(K(E_{p})/ℚ) = [τ]. Since K[ℓ]/H is cyclic

of degree ℓ + 1 and p|ℓ + 1 there is a unique extension H′ of H of degree p contained in K[ℓ]. Let ϕ be any lift of Frob_{ℓ}(H/Q) to Gal(H′/Q). Define

z_{1} = Tr_{K[ℓ]/H′}(y_{ℓ} − ϕy_{ℓ}) − \frac{a_{ℓ}}{p} (y_{H} − ϕy_{H})

Corollary 13.1.6. Suppose ℓ ∤ pD_{K}N and Frob_{ℓ}(K(E_{p})/ℚ) = [τ], and let z_{1} be as above. Then

1. Tr_{H′/H}(z_{1}) = 0 2. For any σ ∈ Gal(H/K), let \overline{σ} denote any lift of σ to Gal(H′/K). Then modulo any λ of

H′ above ℓ,

∑_{σ∈Gal(H/K)} ˜ \overline{σ}z_{1} = −\frac{ℓ + 1 + a_{ℓ}}{p} ỹ.

Lemma 13.1.7. Let E be an elliptic curve over ℚ_{ℓ}. Let E_{1}(\overline{ℚ}_{ℓ}) denote the kernel of the reduction map so there is an exact sequence

0 → E_{1}(\overline{ℚ}_{ℓ}) → E(\overline{ℚ}_{ℓ}) → Ẽ(\overline{𝔽}_{ℓ}).

If m is an integer prime to ℓ then multiplication by m induces an isomorphism

[m] : E_{1}(\overline{ℚ}_{ℓ}) ⟶^{≅} E_{1}(\overline{ℚ}_{ℓ}).

Proof. The proof uses formal groups. See VII.3.1 and IV.3.2 in Silverman’s The Arithmetic of Elliptic Curves.

Lemma 13.1.8. Let A be an abelian variety over a number field K. If A has good reduction at v then

H^{1}(ℚv^{unr}/ℚ_{v}, E(ℚv^{unr}))

is finite, equal to zero for almost all v. In particular, it is zero when A has good reduction at v.

Proof. One proves by reducing modulo v, that H^{1}(ℚv^{unr}/ℚ_{v}, E(ℚv^{unr})) is isomorphic to the cohomology of the component group of the Neron model of A. When A has good reduction the component group is trivial so its cohomology is trivial. For details see Milne’s Arithmetic Duality Theorems, I.3.8.

Proposition 13.1.9. For all but a finite number of p, the following is true. Suppose ℓ ∤ pD_{K}N and Frob_{ℓ}(K(E_{p})/ℚ) = [τ]. Then there is an element c_{ℓ} ∈ H^{1}(ℚ, E)_{p} such that

(1) res_{v}(c_{ℓ}) = 0 for all v ≠ ℓ,

(2) res_{ℓ}(c_{ℓ}) ≠ 0 in H^{1}(ℚ_{ℓ}, E)_{p} if and only if y ∉ pE(K_{ℓ}).

Proof. We may assume p ∤ [H : K]. Assume also that p is large enough so that

H^{1}(ℚv^{unr}/ℚ_{v}, E(ℚv^{unr}))_{p} = 0.

Since K[ℓ]/H is cyclic of order ℓ + 1 and p|ℓ + 1 there is a unique extension H′/H of degree p. Since p ∤ # Gal(H/K) we have

Gal(H′/K) ≅ ℤ/pℤ ⊕ Gal(H/K).

So there is a unique extension K′/K of degree p in K[ℓ], totally ramified at ℓ and unramified at other primes, such that H′ = HK′. Define

z = Tr_{H′/K′}(z_{ℓ}) ∈ E(K′).

By Corollary 13.1.6

Tr_{K′/K}(z) = Tr_{H′/K}(z_{1}) = Tr_{H/K}(Tr_{H′/H}(z_{1})) = Tr_{H/K}(0) = 0.

Fix a generator σ of Gal(K′/K). Since Gal(K′/K) is cyclic there is an isomorphism

Ĥ^{−1}(K′/K, E(K′)) = ker(Tr_{K′/K})/(σ − 1)E(K′) ⟶ H^{1}(K′/K, E(K′))

[Look in chapter IV, section 8 of Cassels-Frohlich for the cohomological properties of cyclic groups.] This map sends z to the cohomology class represented by the cocycle

σ ↦ z, σ^{2} ↦ (1 + σ)z, σ^{3} ↦ (1 + σ + σ^{2})z, , …

13.1. RUBIN, THE WORK OF KOLYVAGIN ON THE ARITHMETIC OF ELLIPTIC CURVES77

Note that these are exactly the conditions forced by the cocycle condition since if f(σ) = z, then f(σ^{2}) = f(σ) + σ(f(σ)) = (1 + σ)z. Define c_{ℓ} to be the image of z under this map.

Next we show that c_{ℓ} ∈ H^{1}(K, E)_{p}^{+}. Choose ϕ in Corollary 13.1.6 so that ϕ|K′ = τ. [I don’t know how to prove this – or even if it is true. Maybe this is possible since H ∩ K′ = K (since H is unramified and K′ is totally ramified at ℓ) and Frob_{ℓ}(H/ℚ)|K = Frob_{ℓ}(K/ℚ) = τ.] Next recall that if G is a group and N is a normal subgroup of G then for any g ∈ G, gN = Ng since gNg^{−1} = N. This means that “automorphisms commute with traces when every extension involved is Galois”. Since ϕ|K′ = τ,

z = Tr_{K[ℓ]/K′}(y_{ℓ}) − τ(Tr_{K[ℓ]/K′}(y_{ℓ})) − \frac{a_{ℓ}}{p} (Tr_{H′/K′}(y_{H}) − τ(Tr_{H′/K′}(y_{H})).

It is clear that τ(z) = −z. We know that τ acts via conjugation as −1 on Gal(K′/K) [why? is this from class field theory?]. Let f be a cocycle representing c_{ℓ}, then

τ ∗ f(σ) = τf(τ^{−1}στ) = τf(σ^{−1}) = −τf(σ) = −τz = z = f(σ)

so c_{ℓ} ∈ H^{1}(K, E)_{p}^{+}.

The Hochschild-Serre spectral sequence gives an exact sequence

0 → H^{1}(K/ℚ, E) → H^{1}(ℚ, E) → H^{1}(K, E)^{+} → H^{2}(K/ℚ, E)

Taking p-torsion and using the fact that p > 2 and that Gal(K/ℚ) Is a group of order two hence its cohomology has trivial p-torsion we obtain an isomorphism

H^{1}(ℚ, E)_{p} ⟶^{≅} H^{1}(K, E)_{p}^{+}.

Thus we may consider c_{ℓ} to be the unique element of H^{1}(ℚ, E) which restricts to c_{ℓ} ∈ H^{1}(K, E). For v ≠ ℓ, since K′/K is unramified at v and p was chosen sufficiently large,

res_{v}(c_{l}) ∈ H^{1}(ℚv^{unr}/ℚ_{v}, E(ℚv^{unr}))_{p} = 0.

To complete the proof of the proposition we must determine the order of res_{l}(c_{l}) in H^{1}(ℚ_{ℓ}, E)_{p}. Write I_{ℓ} = Gal(\overline{ℚ}_{ℓ}/ℚℓ^{unr}) for the inertia subgroup of Gal(\overline{ℚ}_{ℓ}/ℚ_{ℓ}), and consider the maps

H^{1}(ℚ_{ℓ}, E)_{p} → H^{1}(I_{ℓ}, E(\overline{ℚ}_{ℓ}))_{p} → H^{1}(I_{ℓ}, Ẽ(\overline{𝔽}_{ℓ}))_{p} → Hom(I_{ℓ}, Ẽ_{p})

The first map is injective because its kernel, H^{1}(ℚℓ^{unr}/ℚ_{ℓ}, E(ℚℓ^{unr}))_{p} = 0, since E has good reduction at ℓ. The second map is an isomorphism because multiplication by p is an isomorphism on the kernel E_{1}(\overline{ℚ}_{ℓ}) of reduction modulo ℓ. See Proposition 13.1.2 and Lemma 13.1.7. The third map is an isomorphism because I_{ℓ} acts trivially on \overline{𝔽}_{ℓ} hence on Ẽ and a homomorphism of order p must have image in Ẽ_{p}. Since K/ℚ is unramified at ℓ and c_{ℓ} splits over K′ it is easy to see that the image of c_{ℓ} under this sequence of injections is the homomorphism which sends the chosen generator σ of Gal(K′/K) to z̃ in Ẽ(𝔽_{ℓ^{2}}).

Since K′ and H are linearly disjoint in H′ and their compositum is H′ we have

Gal(H′/K) = Gal(H.K′/K) ≅ Gal(H/K) × Gal(K′/K) ≅ Gal(H′/K′) × Gal(K′/K).

In Corollary 13.1.6(b) viewing the lifts \overline{σ} as elements of Gal(H′/K′) and using the fact that z = Tr_{H′/K′}(z_{1}), we see that

z̃ = −\frac{ℓ + 1 + a_{ℓ}}{p} ỹ.

THE FOLLOWING CAUSED ME MUCH EMBARRASSMENT AND IS WRONG: By Lemma 13.1.4 #[Ẽ(𝔽_{ℓ})] is exactly divisible by p since Ẽ(𝔽_{ℓ})_{p} ≅ ℤ/pℤ. Since p|ℓ+1 we conclude that p exactly divides 2(ℓ + 1) + #[Ẽ(𝔽_{ℓ})] = ℓ + 1 + a_{ℓ}. Thus \frac{ℓ+1+a_{ℓ}}{p} is not divisible by p. END EMBARRASSMENT

Since the finite abelian group Ẽ(𝔽_{ℓ^{2}}) contains elements of order p and, by assumption, ỹ can not be divided by p in Ẽ(𝔽_{ℓ^{2}}) we see that the p-part of ỹ must be nonzero. [Consider a decomposition of Ẽ(𝔽_{ℓ^{2}}) as a direct sum of finite cyclic groups. If the p-part of an element is zero then the element is in the image of multiplication by p since multiplication by p acts as an isomorphism on the prime-to-p factors.] Thus z̃ ≠ 0 which completes the proof.

Corollary 13.1.10. Suppose ℓ ∤ pD_{K}N, Frob_{ℓ}(K(E_{p})/ℚ) = [τ], and y ∉ pE(K_{l}). Then

res_{l}(Sel_{p}(E/ℚ)) = 0.

Proof. By Proposition 13.1.9 there exists c_{ℓ} ∈ H^{1}(ℚ, E)_{p} such that for v ≠ ℓ, res_{v}(c_{ℓ}) = 0 and res_{ℓ}(c_{ℓ}) has order p. By Lemma 13.1.4, E(ℚ_{ℓ})_{p} ≅ ℤ/pℤ, so by Proposition 13.1.3 every s_{ℓ} ∈ res_{l}(Sel_{p}(E/ℚ)) is 0. [Recall, this is because ⟨s_{ℓ}, res_{l}(c_{ℓ})⟩ >_{ℓ}= 0, E(ℚ_{ℓ})/pE(ℚ_{ℓ}) = ℤ/pℤ, and res_{l}(c_{ℓ}) has order p.] THIS PROOF IS NOT CLEAR.

Lemma 13.1.11. Assume p𝔤0. Suppose t ∈ H^{1}(K, E_{p})^{±} and that the image of

t̂= res(t) ∈ H^{1}(K(E_{p}), E_{p}) = Hom(Gal(K̄/K(E_{p})), E_{p}(K̄))

is cyclic. Then t = 0.

Proof. t̂ is Gal(K(E_{p})/K) equivariant so for any σ ∈ Gal(K̄/K(E_{p})) and γ ∈ Gal(K(E_{p})/K), γt̂(γ^{−1}σγ) = t̂(σ). Thus γ^{−1}t̂(σ) = t̂(γ^{−1}σγ) so the image of t̂is Gal(K(E_{p})/K)-equivariant. By theorems of Serre for p𝔤0 we must have Im(t̂) = 0. Also, for p𝔤0, we have H^{1}(K(E_{p})/K, E_{p}) = 0 so t = 0.

13.1.6 Proof of Theorem 1

Theorem 13.1.12 (Bump-Friedberg-Hoffstein,Murty-Murty). There exists infinitely many quadratic imaginary fields K such that the corresponding Heegner point y has infinite order.

Theorem 13.1.13 (Mordell-Weil, Serre). For all p sufficiently large the following 3 conditions hold

(1) y ∉ pE(K),

(2) E has no p-isogeny defined over ℚ, and

(3) H^{1}(K(E_{p})/K, E_{p}) = 0.

Proof. A finitely generated abelian group contains no infinitely divisible elements. By the Mordell-Weil theorem E(K) is finitely generated hence (1) is true. Both (2) and (3) hold by theorems of Serre and complex multiplication.

Theorem 13.1.12 supplies us with a quadratic imaginary field K such that y has infinite order. Fix a prime p ∤ N large enough so that it satisfies the three conditions of Theorem 13.1.13. We proceed to prove that Sel_{p}(E/ℚ) = 0. Notice that this will suffice to prove Theorem 13.1.1 because of the exact sequence

0 → E(ℚ)/pE(ℚ) → Sel_{p}(E/ℚ) → � (E/ℚ) → 0.

Fix any

s ∈ Sel_{p}(E/ℚ) ⊂ H^{1}(ℚ, E)_{p}

and let

ŝ = res(s) ∈ H^{1}(K(E_{p}), E_{p}(K̄))
ŷ = res(y) ∈ H^{1}(K(E_{p}), E_{p}(K̄))^{−}.

Fix F/ℚ galois over ℚ such that both ŝ and ŷ split over F, and let G = Gal(F/ℚ).

Lemma 13.1.14. Let G be as above. Then

G = ŝ^{−1}(E_{p}^{−}) ∪ ŷ^{+}(E_{p}^{+}).

Proof. We must show that for any σ ∈ G, either ŝ(σ) ∈ E_{p}^{−} or ŷ(σ) ∈ E_{p}^{+}. Fix σ ∈ G. Choose any prime ℓ ∤ pD_{K}N such that Frob_{ℓ}(F/ℚ) = [στ]. Then

Frob_{ℓ}(K(E_{p})/ℚ) = [στ]|_{K(E_{p})} = [τ]

and (see Lang, Algebraic Number Theory, [X.1])

Frob_{λ}(F/K(E_{p})) = [(στ)^{2}].

Thus

ŝ_{l} = 0⇔ŝ((στ)^{2}) = 0 and y ∈ pE(K_{ℓ})⇔ŷ((στ)^{2}) = 0.

Since s is in the image of restriction, τ ∗ŝ = ŝ. Since τy = τ(y_{K} −σy_{K}) = −y, τ(ŷ) = −ŷ. Thus

(i)

ŝ((στ)^{2}) = ŝ(στστ) = ŝ(σ) + ŝ(τστ) = ŝ(σ) + τŝ(σ) = (1 + τ)ŝ(σ)

(ii)

ŷ((στ)^{2}) = ŷ(σ) + ŷ(τστ) = ŷ(σ) − τŷ(σ) = (1 − τ)(ŷ(σ))

By Corollary 13.1.10 at least one of (i) or (ii) must be zero. Thus either (i) ŝ(σ) = −τŝ(σ) in which case ŝ(σ) ∈ E_{p}^{−}, or (ii) ŷ(σ) = τŷ(σ) in which case ŷ(σ) ∈ E_{p}^{+}.

Lemma 13.1.15. A group can not be the union of two proper subgroups.

Proof. Elementary.

Thus ŝ(G) = E_{p}^{−} or ŷ(G) = E_{p}^{+}. By Lemma 13.1.11, s = 0 or y ∈ pE(K). Since we chose p such that y ∉ pE(K) it follows that s = 0. This completes the proof of Theorem 13.1.1.

********* End Unverified *********

13.2 Gross, Kolyvagin’s work on modular elliptic curves

********* Begin Unverified *********

[Last Modified: 05/20/97]

13.2.1 Overview

Let E/ℚ be a modular elliptic curve, K a suitable quadratic imaginary extension, and y_{K} the corresponding Heegner point. In this paper Gross proves

Theorem 13.2.1. If p is an odd prime such that Gal(ℚ(E_{p})/ℚ) = GL_{2}(ℤ/pℤ) and p does not divide the order of y_{K} in E(K) then Sel_{p}(E/K) is cyclic, generated by δ(y_{K}).

This is stronger than the result proved in Rubin’s paper (above). For example (with some more work) it can be used to prove that the curve y^{2} + y = x^{3} − x of conductor 37 has finite Tate-Shafarevich group. To prove the theorem, Gross constructs, for a certain collection of square-free integers n, cohomology class c(n) ∈ H^{1}(K, E_{p}). In a sense this is orthogonal to what Rubin did in the above paper. Rubin used powers of p in order to deal with “bad” primes but only constructed classes c(ℓ) for ℓ a prime.

13.2.2 Constructing the Euler System

Fix an odd prime p, large enough so that Gal(ℚ(E_{p})/ℚ) = GL_{2}(ℤ/pℤ). Let K be a (suitable) imaginary quadratic extension and denote by D_{K} its discriminant. We consider only square free integers n prime to D_{K}Np and such that all primes factors ℓ|n satisfy the condition

Frob_{ℓ}(K(E_{p})/ℚ) = [τ]

Let K_{n} be the ring class field of conductor n and 𝔾_{n} = Gal(K_{n}/K).

Use the modular parameterization X_{0}(N) → E and the theory of complex multiplication to construct points y_{n} ∈ E(K_{n}). The y_{n} form the “level zero” Euler system of Heegner points. Take the “derivative” of the system of points y_{n} to obtain points P_{n} ∈ E(K_{n}) which are so well behaved that the cohomology classes c(n) to which they give rise can be understood in terms of the properties of P_{1}. The collection c(n) is then the first level Euler system. The c(n) are obtained from the points P_{n} by using the fact that the map res in the following diagram is an isomorphism (for our choice of p), and that the points P_{n} are constructed in exactly the right way so that they must define 𝔾_{n}-invariant elements of E(K_{n})/pE(K_{n}).

0 ↓

H^{1}(K_{n}/K, E)_{p}

↓

0 →  E(K)/pE(K)  ⟶ H^{1}(K, E_{p}) ⟶ H^{1}(K, E)_{p} → 0
↓  ≅↓ res  ↓
0 → (E(K_{n})/pE(K_{n}))^{𝔾_{n}} ⟶ H^{1}(K_{n}, E_{p})^{𝔾_{n}} ⟶ H^{1}(K_{n}, E)_{p}^{𝔾_{n}}

The Galois group G_{n} = Gal(K_{n}/K_{1}) is isomorphic to the product ∏_{ℓ|n} G_{ℓ} where G_{ℓ} = Gal(K_{ℓ}/K_{1}). Define

Tr_{ℓ} = ∑_{σ∈𝔾_{ℓ}} σ ∈ ℤ[𝔾_{n}].

Let a_{ℓ} = ℓ + 1 − #[Ẽ(𝔽_{ℓ})]. The points y_{n} are an Euler system in the sense that the following two axioms are satisfied: Let n = ℓm, then

ESI Tr_{ℓ} y_{n} = a_{ℓ}y_{m} in E(K_{m})

ESII y_{n} = Frob(λ_{m})(y_{m}) (mod λ_{n}) for all primes λ_{n}|λ_{m}|ℓ where λ_{n} is a prime of K_{n}, etc.

Define D_{ℓ} ∈ ℤ[G_{n}] to be a solution to (σ_{ℓ} − 1)D_{ℓ} = ℓ + 1 − Tr_{ℓ}. (See the details section below.) Let D_{n} = ∏_{ℓ|n} D_{ℓ} ∈ ℤ[G_{n}]. Then using the ES axioms for the y_{n} one proves that [D_{n}y_{n}] ∈ (E(K_{n})/pE(K_{n}))^{G_{n}}. To use the res isomorphism in the above diagram we need elements of E(K_{n})/pE(K_{n}) fixed by the larger group 𝔾_{n} = Gal(K_{n}/K). [Is the following construction a special case of cores??]

There is an exact sequence

0 → G_{n} → 𝔾_{n} → Gal(K_{1}/K) → 0.

Let S be a set of coset representatives for G_{n} in 𝔾_{n}, and define

P_{n} = ∑_{σ∈S} σ(D_{n}y_{n}) ∈ E(K_{n})

Because [D_{n}y_{n}] is fixed by G_{n} it follows that [P_{n}] ∈ (E(K_{n})/pE(K_{n}))^{𝔾_{n}}. In light of the above diagram we have constructed the promised cohomology classes c(n) ∈ H^{1}(K, E_{p}). The details are given in the next section.

Details

Proposition 13.2.2. D_{ℓ} exists and is unique up to an integral multiple of ℤ Tr_{ℓ}.

Proof. The existence of D_{ℓ} follows from the more general fact that the polynomial

f(x) = m − (1 + x + ⋯ + x^{m−1}) ∈ ℚ[x]

is divisible by x − 1 [it has 1 as a root], hence by Gauss’ lemma 14.2.1 also divisible by x − 1 in ℤ[x]. For uniqueness up to an element of ℤ Tr_{ℓ} note that ℤ Tr_{ℓ} is the kernel of the map ℤ[G_{ℓ}] ⟶^{σ_{ℓ}−1} ℤ[G_{ℓ}]. [WORRY: maybe D_{ℓ} is as unique as above in ℤ[G_{ℓ}] but is it this unique in ℤ[G_{n}]?]

Theorem 13.2.3. The y_{n} satisfy ESI and ESII.

Proposition 13.2.4.

[D_{n}y_{n}] ∈ (E(K_{n})/pE(K_{n}))^{G_{n}}

Proof. It suffices to show that [D_{n}y_{n}] is fixed by σ_{ℓ}, for all primes ℓ|n, as these elements generate G_{n}. Hence we must prove that (σ_{l} − 1)D_{n}y_{n} lies in pE(K_{n}).

Write n = ℓ · m. We have

(σ_{ℓ} − 1)D_{n} = (σ_{ℓ} − 1)D_{ℓ} · D_{m} = (ℓ + 1 − Tr_{ℓ})D_{m}

in ℤ[G_{n}]. Hence

(σ_{ℓ} − 1)D_{n}y_{n} = (ℓ + 1)D_{m}y_{n} − D_{m}(Tr_{ℓ} y_{n}).

Since ℓ + 1 ≡ 0 (mod p) it suffices to show that Tr_{ℓ} y_{n} lies in pE(K_{m}). This follows from ESI.

13.2.3 Properties

WRITE THIS UP!!! Where do the c(n) live in the eigenspace decomposition H^{1}(K, E_{p})^{+} ⊕ H^{1}(K, E_{p})^{−}?

********* End Unverified *********

13.3 Mccallum, Kolyvagin’s work on Shafarevich-Tate groups

********* Begin Unverified *********

Commutative Algebra

14.1 Basic Definitions

********* Begin Unverified *********

Definition 14.1.1. An element x in a ring R is irreducible if it is not a unit and whenever x = ab with a, b ∈ R then one of a or b must be a unit. An integral domain R is a unique factorization domain (UFD) if every nonzero element of R can be written uniquely (up to units) as a product of irreducible elements of R.

Proposition 14.1.2. A principal ideal domain must be a unique factorization domain.

Theorem 14.1.3. If R is a UFD then the polynomial ring R[x_{1}, …, x_{n}] is a UFD.

Example 14.1.4. The number 6 ∈ ℤ is not irreducible because 6 = 2 · 3 and neither 2 nor 3 is a unit. Also, 1 is not irreducible because 1 is a unit. The polynomial ring k[x] is a UFD because, as the division algorithm shows, it is a principal ideal domain. The ring ℤ[\sqrt{−5}] is not a UFD because

6 = 2 · 3 = (1 + \sqrt{−5})(1 − \sqrt{−5}).

Any field k is (vacuously) a UFD.

********* End Unverified *********

14.2 Basic Facts

********* Begin Unverified *********

Theorem 14.2.1 (Gauss Lemma). Suppose R is a UFD with quotient field K. If f is irreducible in R[x] then f is irreducible in K[x].

Example 14.2.2. If f = x^{2} − x − 1 then f is irreducible in ℤ[\sqrt{5}][x] but reducible in ℚ(\sqrt{5})[x] (because F is reducible iff it has a root and its roots are (1 ± \sqrt{5})/2.) Note that the ring of integers of ℚ(\sqrt{5}) has class number 1, so this example shows that an order in the ring of integers of a field having class number 1 need not be a UFD.

********* End Unverified *********

14.3 Module theory

********* Begin Verified *********

14.3.1 Localization

Nakayama’s lemma can be thought of as a tool which allows one to understand properties of a module over a local ring in terms of properties of its reduction modulo the maximal ideal. This is useful because the reduction is a vector space over the residue field.

Lemma 14.3.1 (Nakayama). Let M be a finitely generated module over a local ring (R, 𝔪). If M ⊗ R/𝔪 = 0 then M = 0.

As a consequence of Nakayama’s lemma we see that if x̄_{1}, …, x̄_{n} generate M ⊗R/𝔪 then any lifting x_{1}, …, x_{n} generates M. Indeed, if N is the quotient of M by the submodule generated by x_{1}, …, x_{n} then N ⊗ R/𝔪 = 0 so N = 0. It follows that the notion of minimal generating set of M is well-defined: all such sets have the same cardinality, and are characterized by the fact that they reduce to a basis of the vector space M ⊗ R/𝔪.

Proposition 14.3.2 (Local-global principal). Let R be a ring and M an R-module. The natural map

0 → M → ∏_{all 𝔪} M_{𝔪}

is an injection (the product is over all maximal ideals 𝔪 of R.) In other words, x = 0 iff x_{𝔪} is 0 in M_{𝔪} for all maximal ideals 𝔪.

Proof. Suppose x ∈ M maps to 0 in the product. Then the annihilator I of x is not contained in any maximal ideal 𝔪 of R so I = R, whence 1 annihilates x and so x = 0.

Proposition 14.3.3. Localization is an exact functor.

Example 14.3.4. If 𝔭 is any prime ideal, then R_{𝔭} is a flat R-module since M_{𝔭} = M ⊗ R_{𝔭}.

14.3.2 Projective, injective, and flat modules

Definition 14.3.5. Let M be an A-module. Then M is flat over A if the functor M ⊗_{A} − is an exact functor on the category of all A-modules. The module M is projective if the functor Hom_{A}(M, −) is exact. The module M is injective if the functor Hom_{A}(−, M) is exact.

An A-module M is finitely presented if it is the cokernel of a morphism of finite rank free modules. A finitely generated module need not be finitely presented. For example, suppose R is a ring and I is an ideal which is not finitely generated. Then the R-module R/I is generated by the image of 1 but it can not be finitely presented as the following proposition demonstrates.

Proposition 14.3.6. The kernel of a surjection T → M → 0 from a finitely generated module T onto a finitely presented module M is finitely generated.

Proof. Bourbaki I.8.9

Theorem 14.3.7. Let M be a finitely presented module over a (not necessarily Noetherian) ring R. Then the following are equivalent:

1. M is flat over R.

2. M_{𝔪} is flat over R_{𝔪} for all maximal ideals 𝔪 of R.

3. M_{𝔪} is a free module over R_{𝔪} for all maximal ideals 𝔪 of R.

4. M_{𝔪} is a projective module over R_{𝔪} for all maximal ideals 𝔪 of R.

5. M is a projective R-module.

Proof. (1⇒2) Suppose 0 → S → T is an injective mapping of R_{𝔪} modules. Viewing S and T as R-modules and tensoring with the flat module M gives an injection

0 → S ⊗_{R} M → T ⊗_{R} M.

But

S ⊗_{R} M ≅ (S ⊗_{R_{𝔪}} R_{𝔪}) ⊗_{R} M ≅ S ⊗_{R_{𝔪}} (R_{𝔪} ⊗_{R} M) ≅ S ⊗_{R_{𝔪}} M_{𝔪}

so

0 → S ⊗_{R_{𝔪}} M_{𝔪} → T ⊗_{R_{𝔪}} M_{𝔪}

is exact and hence M_{𝔪} is flat.

(2⇒1) Suppose 0 → S → T is an injective map of R-modules. Tensor with M and let K be the kernel:

0 → K → S ⊗ M → T ⊗ M.

Localize at a maximal ideal 𝔪 to obtain the exact sequence

0 → K_{𝔪} → S_{𝔪} ⊗ M_{𝔪} → T_{𝔪} ⊗ M_{𝔪}.

Since M_{𝔪} is flat, K_{𝔪} = 0. Since this holds for all maximal ideals 𝔪 it follows by Proposition 14.3.2 that K = 0, hence M is flat.

(2⇒3) Fix a maximal ideal 𝔪. By first localizing at 𝔪 it suffices to prove that if R is a local ring with maximal ideal 𝔪 and M is a flat R-module then M must be free. Thus assume R is local with maximal ideal 𝔪 and that M is flat over R. Let n be the minimal number of generators of M and consider a surjection R^{n} → M → 0. Let K be the kernel so the sequence

0 → K → R^{n} → M → 0

is exact. Tensor with R/𝔪 to obtain the exact sequence

Tor_{1}(M, R/𝔪) ⟶^{f} K ⊗ R/𝔪 ⟶^{g} R^{n} ⊗ R/𝔪 ⟶^{h} M ⊗ R/𝔪 → 0.

Since h is a surjective morphism of vector spaces of the same dimension it must be an isomorphism. Thus g = 0 and f is surjective. But M is flat so Tor_{1}(M, R/𝔪) = 0 and thus K ⊗ R/𝔪 = 0. By Nakayama’s lemma K = 0 so M ≅ R^{n}.

(3⇒4) Free modules are projective.

(4⇒5) This follows from [Eisenbud, Exercise 2.13b] which asserts that a short exact sequence

0 → A → B → C → 0,

with C finitely presented, splits everywhere locally iff it splits globally. (With B free and M = C, use the criterion that a module is projective iff it is the direct summand of a free module.)

(5 ⇒4) If M is projective then M is a direct summand of a free module F. Since M is finitely presented F can be chosen to be finitely generated, F ≅ R^{n}. Then for any maximal ideal 𝔪, M_{𝔪} is a direct summand of the free module R_{𝔪}^{n}.

(4⇒2) We can compute Tor_{1}(M, −) using a projective resolution of M. Since M is projective Tor_{1}(M, −) must vanish. Thus the functor M ⊗ − is exact and so M is flat.

********* End Verified *********

14.4 Étale extensions

********* Begin Verified *********

Definition 14.4.1. A homomorphism A → B of rings is étale if it makes B a flat A-module and if for every prime 𝔓 of B the map A_{𝔭} → B_{𝔓} is unramified, i.e., 𝔭B_{𝔓} = 𝔓B_{𝔓}.

14.5 Local Rings

********* Begin Unverified *********

14.5.1 Regular Local Rings

A local ring R with maximal ideal 𝔪 is regular if 𝔪 can be generated by dim R elements.

Proposition 14.5.1. A regular local ring is a unique factorization domain.

Example 14.5.2. The ring (ℂ[x, y]/(y^{2} − x^{3}))_{(x,y)} is a domain but it is not regular.

********* End Unverified *********

14.6 Examples

********* Begin Unverified *********

Example 14.6.1. Let \overline{ℤ} denoted the integral closure of ℤ in a fixed algebraic closure \overline{ℚ} of ℚ. Then every nonzero prime of \overline{ℤ} is maximal.

Proof. Let 𝔭 ⊂ \overline{ℤ} be a nonzero prime. Any 0 ≠ a ∈ 𝔭 satisfies an integral equation with nonzero constant term hence 𝔭 ∩ ℤ ≠ 0. We can show that 𝔭 is a maximal ideal of \overline{ℤ} by showing that every element of \overline{ℤ} − 𝔭 has an inverse modulo 𝔭. Thus let a ∈ \overline{ℤ} be nonzero. Let O be the ring of integers of the field ℚ(a). Then because 𝔭 ∩ ℤ ≠ 0 and because O is a dedekind domain we know that 𝔭 ∩ O is a nonzero prime ideal of O. Thus there exists b ∈ O such that

ab ≡ 1 (mod 𝔭 ∩ O)

hence ab − 1 ∈ 𝔭 ∩ O ⊂ 𝔭 so ab ≡ 1 (mod 𝔭). Thus a has an inverse modulo 𝔭.

Example 14.6.2. The ring \overline{ℤ} is not noetherian (even though it has Krull dimension 1). For example the chain of ideals

\overline{ℤ}\sqrt{2} ⊂ \overline{ℤ}\sqrt[3]{2} ⊂ \overline{ℤ}\sqrt[4]{2} ⊂ ⋯

is strictly increasing but does not terminate.

********* End Unverified *********

14.7 Associated primes and primary decomposition

Let R be a ring and let M be an R-module. For each element x ∈ M there is an exact sequence

0 → Ann(x) → R → Rx → 0

where Rx ⊂ M is the submodule generated by x and Ann(x) = {r ∈ R : rx = 0}.

Definition 14.7.1. The set Ass(M) of associated primes of the R-module M is the set of primes 𝔭 of R of the form Ann(x) for some x ∈ M.

For “most” x ∈ M the ideal Ann(x) will not be prime.

Example 14.7.2. Construct a scheme X by taking a finite disjoint union of integral affine schemes, thus X = Spec R_{1} ⊕ ⋯ ⊕ R_{n} where each R_{i} is an integral domain. Then the set of associated primes of the zero ideal in R_{1} ⊕ ⋯ ⊕ R_{n} is the set {η_{i} : i = 1, …, n} where η_{i} = R_{1} ⊕ ⋯ ⊕ R̂_{i} ⊕ ⋯ ⊕ R_{n} (R_{i} omitted) is the “ideal sheaf” or “generic point” of the closed subscheme Spec R_{i} ⊂ X.

Example 14.7.3. Let R = k[x, y]/(xy) be the coordinate ring of the union of the x-axis and y-axis in affine two space. The associated primes of the zero ideal of R are (x) and (y), i.e., the ideal sheaves of the two irreducible components.

Example 14.7.4. Let R = k[ε]/ε^{2} be a “fuzzy point”. Then its only associated prime is (ε) which defines the closed subscheme Spec k[ε]/ε.

Example 14.7.5. Let R = k[x, y]/(x^{2}, xy) be a vertical line with a fuzzied origin. The two associated primes are Ann(x) = (x, y) which corresponds to the zero ideal in R, and Ann(y) = (x) which corresponds to the y-axis.

Definition 14.7.6. Let 𝔭 ∈ Spec R and let M be an R-module. The 𝔭-primary submodules of N are those submodules for which Ass(M/N) = {𝔭}.

Theorem 14.7.7 (Primary Decomposition). Let R be a noetherian ring and let M be a finitely generated R-module. Then every proper submodule N ⊂ M has a primary decomposition, i.e., there exists distinct submodules N_{1}, …, N_{r} such that

N = N_{1} ∩ ⋯ ∩ N_{r}

with Ass(M/N_{i}) = {𝔭_{i}}, 𝔭_{i} ∈ Spec R, for each i. Furthermore, Ass(M/N) = {𝔭_{1}, …, 𝔭_{r}}.

For example, the theorem implies that an ideal I in a noetherian ring R has only finitely many associated primes.

Example 14.7.8. Let k be a field and let R = ∏_{i=1}^{∞} k and let M = R. Then M is finitely generated, but the set of associated primes of M is infinite, consisting of products where the ith factor is omitted. Thus R can not be noetherian.

Example 14.7.9. Even in a noetherian ring a single nonzero divisor can lie in infinitely many prime ideals. For example consider R = Spec ℤ[x]. Then x ∈ (x, p) for all primes p and the ideals (x, p) are all maximal.

14.8 Going up and going down

Theorem 14.8.1 (Lying over). Let B be an integral extension of a ring A and let 𝔭 be a prime ideal of A. Then there is a prime 𝔓 of B such that 𝔓 ∩ A = 𝔭. Thus the morphism Spec B → Spec A is surjective.

Theorem 14.8.2 (Going up). Let B be an integral extension of a ring A, suppose 𝔭 ⊂ 𝔭′ are prime ideals of A and that 𝔓 is a prime ideal of B such that 𝔓 ∩ B = 𝔭 then there exists a prime ideal 𝔓′ of B containing 𝔓 and such that 𝔓′ ∩ A = 𝔭′.

Theorem 14.8.3 (Going down). Suppose A is integrally closed and B is a domain, which is integral over A, or that B is flat as an A-algebra. Suppose 𝔭 ⊂ 𝔭′ are prime ideals of A and that 𝔓′ is a prime of B such that A ∩ 𝔓′ = 𝔭′. Then there is a prime ideal 𝔓 of B such that 𝔓 ⊂ 𝔓′ and A ∩ 𝔓 = 𝔭.

The picture I have of going up and going down is as follows. Imagine a morphism between two surfaces (one is floating above the other). Fix a curve on the bottom surface and a point on the curve. Fix a curve on the top surface which maps down to the curve on the bottom surface. Then going up says that there is a point on the top curve which maps down to the point on the bottom curve. An example where this doesn’t hold is the morphism given by projecting the hyperbola xy = 1 onto the x axis y = 0 in affine two space. Note that \frac{1}{x} is not integral over k[x]. Going down says that, upon picking a point on the top surface which maps down to the point on the bottom surface, there is a curve on the top surface which contains the point and maps to the curve on the bottom surface.

14.9 Discriminants and resultants

Definition 14.9.1. Let k be a field and let f ∈ k[x] be a polynomial with distinct roots u_{1}, …, u_{n} ∈ k̄. Then the discriminant of f is Δ^{2} where

Δ = ∏_{i<j}(u_{i} − u_{j}).

Proposition 14.9.2. The discriminant of f ∈ k[x] lies in k.

Proof. Any automorphism leaves the root of f invariant so it can only change the sign of Δ and hence must fix Δ^{2}.

Proposition 14.9.3. The discriminant of x^{3} + ax + b is −4a^{3} − 27b^{2}.

Iwasawa Theory

********* Begin Unverified *********

15.1 The Main Conjecture Of Iwasawa Theory

These are the notes from a talk I gave on the appendix to Rubin. The notation is as in Rubin’s article in the appendix to Lang’s Cyclotomic Fields. The references are to that article. Kevin Buzzard helped me immensely in preparing and I want to thank him here.

15.1.1 Introduction

I am going to talk about the proof of Theorem 7.2. Theorem 7.2 uses the Euler system of cyclotomic units to construct many relations in the Iwasawa algebra Λ. These relations imply that f_{χ}|h_{χ}. The construction involves many different objects including

• fields: K_{n} = ℚ(μ_{p^{n+1}}), F = K_{n}^{+}

• galois groups: G_{n} = Gal(K_{n}/K_{0}), Δ = Gal(K_{0}/ℚ) ≅ (ℤ/pℤ)^{∗}, G_{∞} = Gal(K_{∞}/K_{0}) ≅ ℤ_{p}, Gal(K_{n}/ℚ) ≅ ℤ_{p} × Δ

• characters, projections: χ : Δ → ℤ_{p}^{∗}, e(χ) = \frac{1}{p−1} ∑_{δ∈Δ} χ(δ)^{−1}δ ∈ ℤ_{p}[Δ]

• class groups: C_{n} = {𝔠_{i}, … } = p part of Cl(K_{n})

• Λ-modules: C_{∞} = lim C_{n}, E_{∞}, V_{∞}, Λ = lim ℤ_{p}[G_{n}]

• fractional ideals: J, I_{ℓ}

• an Euler system: ξ_{r}, κ_{r} ∈ F^{×}/(F^{×})^{M}

• positive integers and primes: M, l, p > 2

15.1.2 Recall

Let \overline{E}_{n} denotes the closure in U_{n} of E_{n} ∩ U_{n}, V_{n} the closure of E_{n} ∩ U_{n}. Thus

C_{∞} = lim C_{n}, E_{∞} = lim Ē_{n}, V_{∞} = lim V_{n}

Fix an even character χ : Δ → ℤ_{p}^{∗}. Then C_{∞}(χ) is torsion so by the structure theorem for torsion Λ-modules, there is a quasi-isomorphism

C_{∞}(χ) = ⨁_{i=1,... ,k} Λ/f_{i}Λ

Let f_{χ} = char (C_{∞}(χ)) = ∏ f_{i} and similarly define h_{χ} = char (E_{∞}(χ)/V_{∞}(χ)). I will talk about why f_{χ}|h_{χ}

In order to utilize our Euler system we move to the finite level. To do this reduce modulo I_{n} = (γ^{p^{n+1}} − 1)Λ, where γ is a topological generator of Gal(K_{∞}/K_{0}). Also set

Λ_{n} = Λ/I_{n}Λ = ℤ_{p}[Gal(K_{n}/K_{0})]

15.1.3 The connection between F^{×}/(F^{×})^{M} and the Iwasawa algebra

Fix a positive integer n and an even character

χ : Δ → ℤ_{p}^{∗}.

Let F = K_{n}^{+} be the fixed field of complex conjugation. M denotes a large power of p (how large will be determined later).

To connect the Euler system with the Iwasawa algebra we define a map

\overline{v}_{ℓ} : F^{×}/(F^{×})^{M} → Λ_{n}

Let l ∈ ℚ be a rational prime which splits completely in F and fix a prime ℓ|l. Let I_{ℓ} denote the group of fractional ideals with support in the primes of F lying over l, i.e., the free abelian group on the primes lying over l. For w ∈ F let (w)_{l} denote the projection of the fractional ideal (w) generated by w onto I_{ℓ}. Define

I_{ℓ}(χ) := e(χ)(I_{ℓ} ⊗ ℤ_{p})

Since l splits completely, I_{ℓ} ⊗ ℤ_{p} is free of rank l − 1 over Λ_{n}. Then I_{ℓ}(χ) is free of rank 1 over Λ_{n} generated by ℓ(χ) := e(χ)ℓ. [This is because e(χ)ℓ has precisely one nonzero component in each of the l − 1 Δ orbits of I_{ℓ} ⊗ ℤ_{p} and the coefficients are p-adic integers.] Thus the equation

v_{ℓ}(w)ℓ(χ) = e(χ)(w)_{ℓ}

defines a map

v_{ℓ} : F^{×} → Λ_{n}.

Reducing mod M gives a map

\overline{v}_{ℓ} : F^{×}/(F^{×})^{M} → Λ_{n}/MΛ_{n}

It is by this map that we call our highly controlled κ_{r} ∈ F^{×}/(F^{×})^{M} into play.

15.1.4 The Argument

To show that for every even character χ : Δ → ℤ_{p}^{∗}, that f_{χ}|g_{χ} in Λ we proceed as follows.

1. Fix a special finite index ideal ℂ ⊂ Λ whose existence is guarenteed by the theorems in sections 6.

2. Show that for every η ∈ ℂ (with η relatively prime to γ^{p^{n}} − 1, for all n) that

f_{χ}|η^{k+1}g_{χ} in Λ_{n}/p^{n}Λ_{n}

(Recall that k is the number of factors appearing in the decomposition of C_{∞}(χ).)

3. Using the Krull intersection theorem (see the appendix) we can see that if a|b in Λ_{n}/p^{n}Λ_{n}

for each n then a|b in Λ. Thus

f_{χ}|η^{k+1}g_{χ} in Λ.

4. Because ℂ has finite index it can be shown that ℂ must contain relatively prime elements η and η′. We can choose them prime to γ^{p^{n}} − 1 for all n and then repeating the above steps shows that f_{χ}|(η′)^{k+1}h_{χ} in Λ whence f_{χ}|h_{χ}.

15.1.5 Choosing cunning primes

For our purposes, step 2 is the truly inspired part of the argument. It is accomplished using the Euler system of cyclotomic units. This we now elaborate on.

Fix n. Let h be a positive integer such that

p^{h} ≥ #(Λ_{n}/ηΛ_{n}) and p^{h} ≥ (Λ_{n}/h_{χ}Λ_{n})

Let

M = #(C_{n}(χ))p^{n+(k+1)h}.

Choose 𝔠_{1}, …, 𝔠_{k} ∈ C_{n} so that, for each i, a condition on Ann(𝔠_{i}) ⊂ Λ_{n} of 𝔠_{i} in

C_{n}(χ)/(Λ_{n}𝔠_{1} + ⋯ + Λ_{n}𝔠_{i−1})

is satified. The condition is

ℂAnn(𝔠_{i}) ⊂ f_{i}Λ_{n}

Let 𝔠_{k+1} ∈ C_{n} be any ideal class (the trivial class is ok).

Induction, some Iwasawa theory, and our previous hard work applying the Chebotarev density theorem yields primes ℓ_{i} such that

ℓ_{i} ∈ 𝔠_{i}, l_{i} ≡ 1 (mod M), (ℓ_{i}|l_{i})
\overline{v}_{ℓ_{1}}(κ_{l_{1}}) = u_{1}ηh_{χ}
f_{i−1}\overline{v}_{ℓ_{i}}(κ_{r_{i}}) = u_{i}η\overline{v}_{ℓ_{i−1}}(κ_{r_{i−1}}), i ≥ 2

where u_{i} ∈ (ℤ/Mℤ)^{∗}, and r_{i} = ∏_{j≤i} l_{j}. These relations all take place in Λ_{n}/MΛ_{n}. Staring at this for a moment one sees that

η^{k+1}h_{χ} = u(∏_{i=1}^{k} f_{i})\overline{v}_{ℓ_{k+1}}(κ_{r_{k+1}})

in Λ_{n}/MΛ_{n} and hence in Λ_{n}/p^{n}Λ_{n}. Here is what you see when you stare:

u(∏_{i=1}^{k} f_{i})f_{k}\overline{v}_{ℓ_{k+1}}(κ_{r_{k+1}}) = = u(∏_{i=1}^{k−1} f_{i})u_{i}η\overline{v}_{ℓ_{k}}(κ_{r_{k}})
= ⋯ = u′η^{k}\overline{v}_{ℓ_{1}}(κ_{l_{1}}) = u′′η^{k+1}h_{χ}

How to choose the primes

Step 1: choose ℓ_{1}

We apply theorem 3.1. As input we have

1. ideal class 𝔠_{1}

2. integer M

3. the following map from the finite submodule

W = (E/E^{M})(χ) ⊂ F^{×}/(F^{×})^{M}

to (ℤ/Mℤ)[Gal(F/ℚ)]

ψ : W → Ē(χ)/Ē(χ)^{M} ⟶^{θ} Λ_{n}/MΛ_{n} ⟶^{e(χ)} e(χ)(ℤ/Mℤ)[Gal(F/ℚ)].

Here θ is constructed in Corollary 6.4 using the structure theorem and satisfies

θ(ξ(χ)) = ηh_{χ}

[The existence of such a θ isn’t too surprising because Λ/h_{χ}Λ is closely related to E_{∞}/V_{∞} (essentially units over cyclotomic units) and ξ(χ) is related to V_{∞}.] Theorem 3.1 produces a prime ℓ_{1} ∈ 𝔠_{1} such that l_{1} ≡ 1 (mod M). A computation shows that the other relation is satisfied:

\overline{v}_{ℓ_{1}}(κ_{ℓ_{1}})ℓ_{1}(χ) = e(χ)[κ_{l_{1}}]_{l_{1}} = e(χ)φ_{l}(κ_{1}) = u_{1}ψ(κ_{1})ℓ_{1}(χ)
= u_{1}θ(ξ(χ))ℓ_{1}(χ) = u_{1}ηh_{χ}ℓ_{1}(χ)

Equality 1: By definition. Equality 2: By prop. 2.4 in which we “factored” κ_{l_{1}} (in terms of the map φ_{l_{1}}. Equality 3: By theorem 3.1. Equality 4: By definition of ψ. Equality 5: By construction of θ.

Because (J_{l_{1}}/MJ_{l_{1}})(χ) is free over Λ_{n}/MΛ_{n} this gives the relation.

Step 2: induct to choose ℓ_{i}, 2 ≤ i ≤ k + 1

The induction is continued by replacing W by the submodule W_{i} generated by e(χ)κ_{r_{i−1}}, then using the inductive hypothesis to construct a map ψ_{i} : W_{i} → Λ_{n}/MΛ_{n} with the right properties. Apply theorem 3.1 and check as above that the desired relation is satisfied.

15.1.6 More Is True

Combining the fact that f_{χ}|h_{χ} with an analytic class number formula and some Iwasawa theory shows that for every even character χ,

char (C_{∞}(χ)) = char (E_{∞}(χ)/V_{∞}(χ))

which is the Main Conjecture.

15.1.7 Appendix

Theorem 15.1.1. Let a, b ∈ Λ and suppose a|b in Λ_{n}/p^{n}Λ_{n} for all n. Then a|b in Λ.

Proof. (Kevin Buzzard) View Λ as ℤ_{p}[[T]] so I_{n} = ((1 + T)^{p^{n}} − 1). To show that a|b we show that b ∈ (a). Let J_{n} = (p^{n}, (1 + T)^{p^{n}} − 1) so since Λ_{n} = Λ/I_{n}

Λ_{n}/p^{n}Λ_{n} = Λ/J_{n}.

For each n, since a|b in Λ_{n}/p^{n}Λ_{n}, we have b ∈ (a, J_{n}). Wouldn’t it be nice if we could show that

∩_{n}(a, J_{n}) = (a)

Since Λ is a noetherian local ring so is Λ/(a) (if a is a unit we are done, so assume not). Furthermore J_{n} is contained in the nth power of the maximal ideal 𝔪 of Λ:

(p^{n}, (1 + T)^{p^{n}} − 1) ⊂ (p, T)^{n}

This can be seen by induction by noting (set T = 0 and note that the result is p) that

\frac{(1 + T)^{p^{n}} − 1}{(1 + T)^{p^{n−1}} − 1} = 1 + (1 + T)^{p^{n−1}} + ⋯ + (1 + T)^{(p−1)(p^{n−1})} ∈ (p, T)

Since 𝔪 is the maximal ideal of Λ/(a) the Krull intersection theorem says that ∩(𝔪+(a))^{n} = (a). Thus ∩(J_{n} + (a)) = (a), as required.

Group Theory

********* Begin Unverified *********

16.1 Basics

Proposition 16.1.1. Let G be a group and let H and K be subgroups of G. Then (K : H∩K) = (KH : H), and in particular (K : H ∩ K)|(G : H).

Proof. The natural map of (left) coset spaces K/(H ∩ K) → G/H is injective. The image is the collection of cosets KH/H. Thus the cardinality of the coset space K/(H ∩ K) equals the index (KH : H). Since (G : H) = (G : KH)(KH : H) the final assertion follows.

********* End Unverified *********

Definition 16.1.2. A group is simple if it has no proper normal subgroups.

Proposition 16.1.3. Any index 2 subgroup H of a group G is normal.

Proof. 1. If x ∈ G, x ∉ H, then G = H ∪ xH = H ∪ Hx (disjoint union) so xH = Hx, as required.

********* Begin Unverified *********

Definition 16.1.4. Let N and H be groups and let φ : H → Aut(N) be a homomorphism. Then the semidirect product N⋊H (with respect to φ) is the set N × H equipped with the binary operation

(n_{1}, h_{1}) · (n_{2}, h_{2}) = (n_{1}φ(h_{1})(n_{2}), h_{1}h_{2}).

There is an exact sequence

0 → N → N⋊H ⟶^{π_{H}} H → 0.

Example 16.1.5. The dihedral group D_{2n} of order 2n is the semidirect product ℤ/nℤ⋊ℤ/2ℤ with respect to the nontrivial map ℤ/2ℤ → Aut(ℤ/nℤ) = (ℤ/nℤ)^{×} which sends the generator 1 of ℤ/2ℤ to the automorphism −1.

********* End Unverified *********

********* Begin Unverified *********

16.1.1 Characteristic Subgroups

A subgroup H of a group G is a characteristic subgroup if for every automorphism φ of G, φ(H) = H. Any characteristic subgroup of G is normal.

Proposition 16.1.6. If K is a characteristic subgroup of H and H is a characteristic subgroup of G then K is a characteristic subgroup of G.

Proof. An automorphism of G restricts to an automorphism of H.

If K is normal in H and H is normal in G then K need not be normal in G. For example, let G = A_{4}, H = V_{4} be the Klein four group, and K be any two element subgroup of H.

********* End Unverified *********

16.2 Group Actions

********* Begin Unverified *********

Theorem 16.2.1 (Orbit-Stabilizer Theorem). Let G be a finite group, X a finite G-set, and suppose x ∈ X. If H is the subgroup fixing x then [G] = [Gx][H].

********* End Unverified *********

16.3 The Sylow Theorems

********* Begin Verified *********

Let p be a prime number. Let G be a finite group with [G] = p^{n}m and (p, m) = 1. A subgroup H of G is a Sylow p-subgroup if [H] = p^{n}.

Theorem 16.3.1 (Sylow’s Theorems). Let G be a finite group and p a prime dividing [G]. Then

(i) There exists a Sylow p-subgroup of G.

(ii) If H is a p-subgroup of G, then H is contained in some Sylow p-subgroup.

(iii) All Sylow p-subgroups are conjugate.

(iv) The number of Sylow p-subgroups of G is congruent to 1 modulo p.

********* End Verified *********

********* Begin Unverified *********

This proof [1] is by a repeated clever application of the orbit-stabilizer theorem.

Proof. Suppose [G] = p^{n}m with (p, m) = 1. Left multiplication makes the collection P of subsets of G of order p^{n} a G-set. Suppose there is an orbit O ⊂ P such that p ∤ [O]. By multiplying an element of O by an appropriate elements of G we obtain X ∈ O such that 1 ∈ X. If P is the stabilizer of X, then P X = X, and so P ⊂ X hence [P] ≤ [X] = p^{n}. The orbit-stabilizer theorem insures us that [G] = [H][O] so because p ∤ [O] it must be the case that p^{n}|[P]. Thus P = X, P is a Sylow p-subgroup of G, and O is the coset space G/P. Conversely, suppose that P is a Sylow p-subgroup of G. Then the orbit of P in P is the left coset space G/P which has cardinality m which is prime to p. Thus there is a bijection between Sylow p-subgroups of G and orbits in P of cardinality prime to p.

Let P′ be the subset of elements of P which lie in an orbit whose cardinality is not divisible by p. Every orbit of P − P′ has order divisible by p so p|[P − P′] = [P] − [P′]. Thus [P] ≡ [P′] (mod p). As noted above if S ∈ P′ then the orbit of S is the set of cosets of a Sylow p-subgroup of G and hence has cardinality m. Thus

rm = [P′] ≡ [P] = (\begin{matrix}p^{n}m \\ p^{n} \end{matrix}) (mod p).

Since p ∤ m, this implies that the value of r, modulo p, depends only on the order of G and not on G itself; that is, any two groups of the same order have the same number, modulo p, of Sylow p-subgroups. The cyclic group of order p^{n}m has exactly one Sylow p-subgroup so r ≡ 1 (mod p) which proves (iv). In particular, r > 0 which proves (i).

Let P be any Sylow p-subgroup of G and let Q be an arbitrary p-subgroup of G. Then Q acts on the set Y of conjugates of P in G. The cardinality of each Q-orbit of Y is a power of p. We have

[Y] = [G : N_{G}(P)]|[G : N_{G}(P)][N_{G}(P) : P] = [G : P] = m

where N_{G}(P) is the subgroup of G leaving P stable under conjugation. Thus p ∤ [Y] and so there must be at least one orbit of Y which contains exactly one element.

Let P_{1} be an element belonging to a one element Q-orbit of Y. Then for all x ∈ Q, we have xP_{1} = P_{1}x^{−1}; consequently QP_{1} = P_{1}Q and so the set QP_{1} = {xy : x ∈ Q, y ∈ P_{1}} is a subgroup of G. Clearly [P_{1}] ≤ [QP_{1}]; but [QP_{1}] = [Q][P_{1}]/[Q ∩ P_{1}] so QP_{1} must be a p-group. Thus [QP_{1}] = [P_{1}] hence Q ⊂ P_{1}. Since P_{1} is a Sylow p-subgroup this proves (ii). If Q is a Sylow p-subgroup then Q = P_{1} so Q is conjugate to P. This proves (iii).

********* End Unverified *********

********* Begin Unverified *********

Example 16.3.2. Let G = S_{3} be the symmetric group of order 6 = 2 · 3. The Sylow 2-subgroups of G are

⟨(1, 2)⟩, ⟨(1, 3)⟩, ⟨(2, 3)⟩.

Evidently they are all conjugate, there are 3 of them, and 3 ≡ 1 (mod 2). The Sylow 3-subgroup is ⟨(1, 2, 3)⟩.

Let G = S_{4} be the symmetric group of order 24 = 2^{3} · 3. The Sylow 2-subgroups of G are each dihedral of order 8 and are

⟨(3, 4), (1, 2), (1, 3)(2, 4)⟩, ⟨(2, 3), (1, 4), (1, 2)(3, 4)⟩, ⟨(2, 4), (1, 3), (1, 4)(2, 3)⟩.

Again, the number of Sylow 2-subgroups is equivalent to 1 modulo 2. (Note that 3 ≢ 1 (mod 8).) The four Sylow 3-subgroups are

⟨(1, 2, 3)⟩, ⟨(1, 3, 4)⟩, ⟨(1, 2, 4)⟩, ⟨(2, 3, 4)⟩.

********* End Unverified *********

16.3.1 p-groups

********* Begin Unverified *********

Definition 16.3.3. A group G is a p-group if every element of G has order a power of p.

Proposition 16.3.4. A nontrivial finite p-group G has nontrivial center.

Proof. Consider the orbits of G acting on itself by conjugation. If G has center {1} then since each nontrivial orbit has cardinality divisible by p we have 1 ≡ [G] (mod p). This is impossible because [G] is a nonzero power of p.

16.4 Composition Series

********* Begin Unverified *********

Definition 16.4.1. A composition series for a group G is a series of subgroups

G = G_{0} ⊃ G_{1} ⊃ ⋯ ⊃ G_{n} = 1

such that G_{i+1} is normal in G_{i} and the quotients G_{i}/G_{i+1} are simple. The quotients G_{i}/G_{i+1} are the composition factors.

Proposition 16.4.2. Every finite group G has a composition series.

Proof. Since G is finite it contains a maximal normal subgroup G_{1}. But G/G_{1} is simple and, by induction, G_{1} has a composition series; thus G has a composition series.

Theorem 16.4.3 (Jordan-Hölder). Any two composition series for G have, up to permutation, the same composition factors.

For a nice proof see [1].

********* End Unverified *********

16.5 Solvable Groups

********* Begin Unverified *********

Let G be a finite group. If S and T are subsets of G let [S, T] denote the subgroup generated by the set {[s, t] = sts^{−1}t^{−1} : s ∈ S, t ∈ T }. The i-th derived subgroup of G is

G^{(i)} = [G^{(i−1)}, G^{(i−1)}], G^{(0)} = G.

The derived subgroups G^{(i)} are characteristic subgroups of G.

Definition 16.5.1. A finite group G is solvable if there exists a nonnegative integer n such that G^{(n)} = 0.

The group A_{5} is simple and non-abelian so A_{5}^{(i)} = A_{5} for all i. If k is a field containing a non-trivial n-th root of unity, then SL_{n}(k) equals its derived subgroup but has non-trivial center (except when n = 2 and [k] = 3.) Thus a group equal to its derived subgroup need not be simple. If G is both simple and solvable then G is cyclic of prime order.

Proposition 16.5.2. If G is an extension of a solvable group by a solvable group then G is solvable.

Proof. G fits into an exact sequence

0 → H → G → K → 0

with both H and K solvable. Thus the quotient of G by a solvable subgroup is solvable hence the series G^{(i)} must descend to zero after n + m steps where K^{(n)} = H^{(m)} = 0.

Example 16.5.3. The dihedral group D_{2n} of order 2n is solvable. The cyclic subgroup R of rotations in D_{2n} is of index 2 hence normal so there is an exact sequence

0 → ℤ/nℤ → D_{2n} → ℤ/2ℤ → 0.

Since ℤ/nℤ and ℤ/2ℤ are both solvable it follows that D is solvable.

A subnormal series for G is a series

G = G_{0} ⊃ G_{1} ⊃ ⋯ ⊃ G_{n} = 1

in which G_{i+1} is normal in G_{i} for each i. Proposition 16.5.4. G is solvable iff it admits a subnormal series in which the successive quotients G_{i}/G_{i+1} are abelian. Proposition 16.5.5. A group having a composition series is solvable iff all of its composition factors are prime cyclic. Proof. If the composition factors are prime cyclic they are abelian so the previous proposition applies. If the group G is solvable then the composition factors, being quotients of subgroups of G, must themselves be solvable. They are thus simple and solvable hence prime cyclic. Because finite groups have composition series and a simple p-group is prime cyclic (since the center is nontrivial) we obtain the following corollary. Corollary 16.5.6. Finite p-groups are solvable.

16.5.1 Theorems

Theorem 16.5.7. Suppose G is a finite solvable group of order mn with (m, n) = 1. Then G has a subgroup of order m. Furthermore, any two subgroups of order m are conjugate, and any subgroup whose order divides m is contained in a subgroup of order m. Theorem 16.5.8 (Burnside). If p and q are primes, then any group of order p^{a}q^{b} is solvable. Theorem 16.5.9 (Feit-Thompson). All groups of odd order are solvable. Theorem 16.5.10. Let G be a finite group. If G has a subgroup of order m whenever m and n are coprime numbers such that [G] = mn, then G is solvable.

********* End Unverified *********

16.6 Nilpotent Groups

Definition 16.6.1. A group G is nilpotent if there is a normal series

G = G_{0} ⊃ G_{1} ⊃ ⋯ ⊃ G_{n} = 1

in which each quotient G_{i}/G_{i+1} is contained in the center of G/G_{i+1}. Such a series is called a central series. Proposition 16.6.2. Nilpotent groups are solvable. Proposition 16.6.3. Any finite p-group G is nilpotent. Proof. We induct on [G], the case when [G] = p being trivial. The center Z of G is nontrivial so, by induction, G/Z has a central series

G/Z = G_{0}/Z ⊃ G_{1}/Z ⊃ ⋯ ⊃ G_{r}/Z.

The series

G = G_{0} ⊃ G_{1} ⊃ ⋯ ⊃ G_{r} ⊃ G_{r+1} = Z ⊃ 1

is a central series of G. Theorem 16.6.4. The following are equivalent for a finite group G.

• G is nilpotent.

• Every Sylow subgroup of G is normal in G.

• G is the direct product of its Sylow subgroups.

• Every maximal subgroup of G is normal in G.

Group Cohomology

********* Begin Unverified *********

17.1 The Definition of Group Cohomology

Fix a nonnegative integer q. Let G be a finite group and M a G-module. Thus M is a module over the integral group ring ℤ[G]. Give ℤ the trivial G-action and define the cohomology group

H^{q}(G, M) =^{df} Ext_{ℤ[G]}^{q}(ℤ, M).

Since Ext_{ℤ[G]}^{q}(ℤ, −) is the q-th derived functor of Hom_{ℤ[G]}(ℤ, −) and Hom_{ℤ[G]}(ℤ, M) = M^{G} (points fixed by G) it follows that H^{q}(G, −) is the q-th right derived functor of the fixed point functor.

By explicitly choosing a free resolution of ℤ in the category of ℤ[G]-modules one obtains a construction of H^{q}(G, M) as a quotient of cocycles modulo coboundaries. Let P_{i} = ℤ[G^{⊕i+1}], then the standard free resolution of ℤ is

⋯ → P_{2} → P_{1} → P_{0} → ℤ → 0.

Let G′ be any finite group. Then any homomorphism φ : G′ → G induces, for each i, a map P′_{i} → P_{i}. These maps form a morphism of complexes P′_{•} → P_{•}. For any G-module M, applying the contravariant functor Hom(−, M) to each complex and taking cohomology yields the change of groups map H^{q}(G, M) → H^{q}(G′, M).

17.2 Morphisms of Pairs

[Attrib.: Everything in this section was shown to me by Hendrik.]

17.2.1 The category P of pairs

The objects of P are pairs (G, M) where G is a finite group and M is a discrete G-module. A morphism of pairs (G, M) → (G′, M′) is a pair (φ, f) where φ : G′ → G is a group homomorphism and f : M → M′ is a homomorphism of modules such that the following condition is satisfied. For any σ ∈ G′ and m ∈ M, we require that f(φ(σ).m) = σ.f(m). Thus f is a morphism of G′-modules if M is given the structure of G′-module via the map φ : G′ → G. The definition is illustrated in the diagram below.

\begin{matrix}G & ⟵^{φ} & G′ \\ | & & | & f(φ(σ).m) = σ.f(m) \\ M & ⟶^{f} & M′ \end{matrix}

If (φ, f) : (G, M) → (G′, M′) is a morphism of pairs, then φ : G′ → G induces the change of groups map H^{q}(G, M) → H^{q}(G′, M). Viewing M as a G′-module, f induces a morphism H^{q}(G′, M) → H^{q}(G′, M′). Composing these two maps yields a map H^{q}(G, M) → H^{q}(G′, M′). Thus cohomology is a functor

P → Ab (G, M) ↦ H^{q}(G, M).

[Questions: How can I state Shapiro’s Lemma so that I can understand it? Is homology a contravariant functor P → Ab? Are there any undue subtleties when G is profinite?]

17.2.2 Inf, Res, Inner Automorphisms, Shapiro’s Lemma

Let (G, M) ∈ P and let H be a subgroup of G. The morphism of pairs

\begin{matrix}G & ↩ & H \\ | & & | \\ M & ⟶^{id} & M \end{matrix}

induces the restriction map

H^{q}(G, M) ⟶^{res_{H}^{G}} H^{q}(H, M).

Next suppose H is a normal subgroup of G. Then the morphism of pairs

G/H ⟵^{π} G
|  |
M^{H} ↪ M

induces the inflation map

H^{q}(G/H, M^{H}) ⟶^{inf} H^{q}(G, M).

Now suppose σ ∈ G, then conjugation by σ induces an inner automorphism φ_{σ} : G → G. The morphism of pairs

G ⟵^{φ_{σ}} G
|  |
M ⟶^{σ^{−1}} M

induces the identity map on H^{q}(G, M). [See page 99-100 of Cassels-Frohlich for the proof which inductively employs dimension shifting.]

17.2.3 Bimodules

Let R and S be not necessarily commutative rings. Let A be an additive abelian group. A left R-module structure on A is a bilinear map R × A → A such that r′(ra) = (r′r)a for all r, r′ ∈ R and a ∈ A. A right S-module structure on A is a bilinear map A × S → A such that (as)s′ = a(ss′) for all s, s′ ∈ S and a ∈ A. An R-S-bimodule structure on A is a compatible structure of left R-module and right S-module, the compatibility condition being that (ra)s = r(as) for all r ∈ R, a ∈ A, and s ∈ S.

Let R, S, T be rings, A an R-S-bimodule and B an S-T -bimodule. Then the tensor product A ⊗_{S} B is an R-T -bimodule. A ⊗_{S} B is the quotient of A ⊗ B (tensor product of abelian groups) by the relations generated by as ⊗ b = a ⊗ sb. The action is r(a ⊗ b) = (ra) ⊗ b and (a ⊗ b)t = a ⊗ (bt).

Now suppose A is an R-S-bimodule and B is an R-T -bimodule. Then _{R} Hom(A, B), the group of left R-module homomorphisms A → B, is an S-T -bimodule. If φ ∈ _{R} Hom(A, B) then s.φ(a) = φ(as). This is again a left R-module homomorphism since

r(s.φ(a)) = rφ(as) = φ(ras) = s.φ(ra).

17.3. SYLOW P -SUBGROUPS AND SHARPNESS 99

Further, (φ.t)(a) = φ(a)t. Then φ.t is again a left R-module homomorphism since

r(φ.t(a)) = rφ(a)t = φ(ra)t = φ.t(ra).

If A is an S-R-bimodule and B is a T -R-bimodule then the group Hom_{R}(A, B) of right Rmodule homomorphisms A → B is a T -S-bimodule. If φ ∈ Hom_{R}(A, B) then (t.φ)(a) = tφ(a). Then t.φ is a right R-module homomorphism since

(t.φ)(a)r = (tφ(a))r = t(φ(a)r) = tφ(ar) = (t.φ)(ar).

Also, (φ.s)(a) = φ(sa) and φ.s is a right R-module homomorphism since

(φ.s)(a)r = φ(sa)r = φ((sa)r) = φ(s(ar)) = (φ.s)(ar).

In summary we have

RA_{S} ⊗_{S} _{S}B_{T} = _{R}(A ⊗_{S} B)_{T}
R Hom(_{R}A_{S},_{R}B_{T}) = S-T -bimodule
Hom_{R}(_{S}A_{R},_{T} B_{R}) = T -S-bimodule

17.2.4 Shapiro’s Lemma

17.3 Sylow p-subgroups and Sharpness

If a class c ∈ H^{q}(G, M) splits in a subgroup H of G then it must also split in each of the conjugates of H.

Proposition 17.3.1. Let H be a subgroup of G. Then for every σ ∈ G,

ker(res_{H}^{G}) = ker(res_{σ^{−1}Hσ}^{G}).

Proof. The diagram

(G, M)  ⟶^{(φ_{σ},σ^{−1})} (G, M)
↓  ↓
(σHσ^{−1}, M) ⟶^{(φ_{σ},σ^{−1})} (H, M)

commutes in the category P. (The vertical maps are induced by the natural inclusions.) Taking cohomology yields

H^{q}(G, M) ⟶^{id} H^{q}(G, M)
↓ res_{σHσ^{−1}}^{G}  ↓ res_{H}^{G}
H^{q}(σHσ^{−1}, M) ⟶^{≅} H^{q}(H, M)

from which the proposition follows.

If c ∈ H^{q}(G, M), then

H_{c} = {H ⊂ G : res_{H}^{G}(c) = 0}

is closed under conjugation, and contains the subset

P_{c} = {H ∈ H_{c} : #[H] is a prime power}.

Lemma 17.3.2. Let H be a subgroup of G. Then

[G : H] = gcd{[G : H′] : H′ ⊂ H and #[H′] is a prime power }.

Proof. For each prime p|#[H] let H_{p} ⊂ H be a Sylow p-subgroup. Then

[G : H_{p}] = [G : H][H : H_{p}]

so it suffices to note that gcd_{p|#[H]}[H : H_{p}] = 1. [The gcd divides #[H], if p|#[H] then p does not divide [H : H_{p}].] Proposition 17.3.3.

index(c) = gcd_{H∈H_{c}}[G : H] = gcd_{H∈P_{c}}[G : H]

Proof. The first equality is the definition of index. The second equality follows from the above lemma.

Fix, for each prime p, a Sylow p-group G_{p} ⊂ G.

Proposition 17.3.4.

index(c) = gcd_{H∈P′_{c}}[G : H] = ∏_{p} gcd_{\begin{matrix}H⊂G_{p} \\ H∈H_{c} \end{matrix}}[G_{p} : H]

Proof. The first equality holds because, for each p, the Sylow p-subgroups of G are conjugate. The second equality is true because the power of p which divides the first gcd is completely determined by the indexes of the elements H ∈ P_{c} of order a power of p, and the product on the right just combines this information a prime at a time.

Theorem 17.3.5. Suppose the finite group G is a direct sum of its Sylow p-subgroups. Then every G-module is sharp. That is, for every q > 0 every class c ∈ H^{q}(G, M) is sharp.

Proof. If G is a p-group then the result is clear. Now fix c ∈ H^{q}(G, M). By the above proposition we can find, for each prime p, a subgroup H_{p} ⊂ G_{p} which splits c, such that

index(c) = ∏_{p}[G_{p} : H_{p}].

Since G = ∏ G_{p}, the subgroup H = ∏ H_{p} ⊂ G splits c and has index equal to the index of c.

********* End Unverified *********

17.4 The case G ⊂ AutM

********* Begin Unverified *********

Let m be a positive integer and let M be a free ℤ/mℤ module of finite rank. Our goal is to see for which G ⊂ Aut M we have H^{q}(G, M) = 0. First suppose G = Aut M, then (ℤ/mℤ)^{∗} ⊂ G. The morphism of pairs (G, M) → (G, M) corresponding to the inner automorphism determined by a ∈ (ℤ/mℤ)^{∗} induces the identity map H^{q}(G, M) → H^{q}(G, M) (for all q ≥ 0). On the other hand, since a ∈ Z(G) it also induces the multiplication by a map. Thus the integer a − 1 kills H^{q}(G, M). For example, taking a = −1 ∈ (ℤ/mℤ)^{∗} we see that 2 annihilates H^{q}(G, M), so if m is prime to 2 then H^{q}(G, M) = 0.

Next we consider the case when G ⊂ Aut M. From the above discussion we obtain the following proposition.

Proposition 17.4.1. Let M be a free ℤ/mℤ module and let G ⊂ Aut M. If

(a − 1 : a ∈ G ∩ (ℤ/mℤ)^{∗}) = (1)

as ideals in ℤ/mℤ, then H^{q}(G, M) = 0 for all q ≥ 0.

Corollary 17.4.2. Let M be a finite rank free ℤ/mℤ module and let G ⊂ Aut M. If there are no primes p|m such that p − 1|[Aut M : G] then H^{q}(G, M) = 0 for all q ≥ 0.

Proof. Let H = G ∩ (ℤ/mℤ)^{∗}. We must show that (a − 1 : a ∈ H) = (1). Suppose not, then all a − 1 are contained in a common maximal ideal, say (p) with p|m. Consequently, p|a − 1 for each a ∈ H so

H ⊂ ker((ℤ/mℤ)^{∗} → (ℤ/pℤ)^{∗}).

It follows that

p − 1|[(ℤ/mℤ)^{∗} : H] = [Aut M ∩ (ℤ/mℤ)^{∗} : G ∩ (ℤ/mℤ)^{∗}]|[Aut M : G].

Example 17.4.3. The converse of the proposition is not true. For example take m = 2. The converse of the corollary is also false. If m = 6 and G = Aut M then (a−1 : a ∈ (ℤ/mℤ)^{∗}) = (1) so H^{q}(G, M) = 0 for all q ≥ 0, but 2|m and 2 − 1 = 1|[Aut M : G].

17.4.1 Application to galois representations

Theorem 17.4.4. Let A be an abelian variety over a field F and let p ≥ 5 be a prime number such that

Gal(F (A_{p})/F) ≅ Aut(A_{p}).

Then for every quadratic extension K of F and for every integer q ≥ 0 we have

H^{q}(K(A_{p})/K, A_{p}) = 0.

Proof. If K ∩ F (A_{p}) = F then since both F (A_{p}) and K are galois,

Gal(K(A_{p})/K) ≅ Gal(F (A_{p})/F)

and the action on A_{p} is the same. Thus for q ≥ 0,

H^{q}(K(A_{p})/K, A_{p}) = H^{q}(Aut(A_{p}), A_{p}) = 0

since p ≠ 2 and A_{p} is a free ℤ/pℤ module.

If K ∩ F (A_{p}) = K (i.e., K ⊂ F (A_{p})) then Gal(K(A_{p})/K) = Gal(F (A_{p})/K) is an index 2 subgroup of Gal(F (A_{p})/F) = Aut(A_{p}). Thus the index of Gal(F (A_{p})/K) in (ℤ/pℤ)^{∗} is ≤ 2 so since p ≥ 5 there must be a scalar a ∈ Gal(F (A_{p})/K) ⊂ Aut(A_{p}) such that a − 1 is a unit mod p. It follows by the proposition that H^{q}(F (A_{p})/K, A_{p}) = 0 for all q ≥ 0.

Example 17.4.5. Let E be the elliptic curve y^{2} + y = x^{3} − x. Serre showed that for all primes p,

Gal(ℚ(E_{p})/ℚ) = Aut(E_{p}).

It follows from the theorem that for every prime p ≥ 5 and every quadratic extension K/ℚ that

H^{q}(K(E_{p})/K, E_{p}) = 0.

When p = 2 and K ⊂ ℚ(E_{p}), then Gal(ℚ(E_{p})/ℚ) has order 6 so Gal(K(E_{p})/K) has order 3 so since E_{p} is a 2-group H^{q}(K(E_{p})/K, E_{p}) = 0 for all q ≥ 0.

********* End Unverified *********

17.5 Group extensions

English

18.1 Spelling

********* Begin Unverified *********

If ever I misspell a word, it goes here so that I will know not to misspell it again.

arguing behavior contravariant definitely determinants disembodied eliminated embarrassment explicitly feel fertilized gluing happened isogenous judgment kernel lying misspell parabola parallel parameterize pontrjagin preparatory secretly separable similar surprised upcoming using zeroth

********* End Unverified *********

18.2 Grammar

If ever I discover an error in my use of grammar, or learn a handy rule, then it goes here.

Chapter 19

Elementary Number Theory

********* Begin Unverified *********

19.0.1 Density of square free integers

Landau calculated the density of squarefree integers in Primzahlen. Theorem 19.0.1. The number of squarefree integers less than x is

\frac{6}{π^{2}} x + O(\sqrt{x}).

A slight refinement of Landua’s result is contained in [10]. ********* End Unverified *********

Chapter 20

Analysis and Measure Theory

This chapter will contain the basic facts about analysis and measure theory that I find useful.

Bibliography

[1] J.L. Alperin, Rowen B. Bell, Groups and Representations, GTM 162, Spring-Verlag, 1995.

[2] S. Arno, M.L. Robinson, F.S. Wheeler, Imaginary quadratic fields with small odd class number, Algebraic Number Theory preprint server, 1993.

[3] J.W.S. Cassels, Arithmetic on Curves of Genus 1 (V). Two Counter-Examples, Journal London Math. Soc., 38, 244–248, 1963.

[4] Cassels-Frohlich, Algebraic Number Theory, Thompson Book Company, 1967.

[5] R. Hartshorne, Algebraic Geometry, GTM 52, Springer-Verlag, 1977.

[6] S. Lang, Algebraic Number Theory, GTM 110, Springer-Verlag, 2nd edition, 1994.

[7] Stephen Lichtenbaum, Duality Theorems for Curves over P -adic Fields, Inventiones math. 7, 120–136, 1969.

[8] H. Matsumura, Commutative ring theory, Cambridge University Press, 8, 1992.

[9] J.F. Mestra,Construction of an elliptic curve of rank 12, Comptes. Rendus, 295, 643–644, 1982.

[10] K. Rogers, The Schnirelmann density of the squarefree integers, Proceedings of the AMS, 15, 515–516, 1964.

[11] J. Silverman, The Arithmetic of Elliptic Curves, GTM 106, Springer-Verlag, 1986.

[12] J. Silverman, Advanced Topics in the Arithmetic of Elliptic Curves, GTM 151, Springer-Verlag, 1994.

[13] Zariski-Samuel, Commutative Algebra, Van Nostrand, 1963.