2016-02-29-modabvar.sagews

February 29, 2016: Modular abelian varieties

William Stein

Rough plan for rest of the class

• Feb 29: computing with modabvars

• Mar 2 : modular symbols

• Mar 4 : BSD (1/2) (Birch's motivation: modular symbols to compute L(E,1)/Omega exactly)

• Mar 7 : BSD (2/2)

• Mar 9 : p-adic BSD (1/2)

• Mar 11: p-adic BSD (2/2)

Example: Level 43

S = CuspForms(43); S

Cuspidal subspace of dimension 3 of Modular Forms space of dimension 4 for Congruence Subgroup Gamma0(43) of weight 2 over Rational Field

S.dimension()

3

X = S.newforms(names='a')
f0 = X[0]
print "f0=", f0
f1 = X[1]   # there are two newforms associated to this: f1 and its image under Galois!
print "f1=", f1

f0= q - 2*q^2 - 2*q^3 + 2*q^4 - 4*q^5 + O(q^6) f1= q + a1*q^2 - a1*q^3 + (-a1 + 2)*q^5 + O(q^6)

show(f0.q_expansion(20))   # related to an elliptic curve of conductor 43

$\displaystyle q - 2q^{2} - 2q^{3} + 2q^{4} - 4q^{5} + 4q^{6} + q^{9} + 8q^{10} + 3q^{11} - 4q^{12} - 5q^{13} + 8q^{15} - 4q^{16} - 3q^{17} - 2q^{18} - 2q^{19} + O(q^{20})$

show(EllipticCurve('43a').q_eigenform(20))

$\displaystyle q - 2q^{2} - 2q^{3} + 2q^{4} - 4q^{5} + 4q^{6} + q^{9} + 8q^{10} + 3q^{11} - 4q^{12} - 5q^{13} + 8q^{15} - 4q^{16} - 3q^{17} - 2q^{18} - 2q^{19} + O(q^{20})$

show(f1.q_expansion(15))

$\displaystyle q + a_{1}q^{2} - a_{1}q^{3} + \left(-a_{1} + 2\right)q^{5} - 2q^{6} + \left(a_{1} - 2\right)q^{7} - 2 a_{1}q^{8} - q^{9} + \left(2 a_{1} - 2\right)q^{10} + \left(2 a_{1} - 1\right)q^{11} + \left(2 a_{1} + 1\right)q^{13} + \left(-2 a_{1} + 2\right)q^{14} + O(q^{15})$

f1.base_ring()

Number Field in a1 with defining polynomial x^2 - 2

Theorem of Shimura: There is an abelian variety $A_{f_1}$ over $\QQ$ attached to $f_1$ of dimension $2$.

It's endomorphism ring is $\ZZ[\sqrt{2}]$... It's not the Jacobian of a curve...

(Sketch, on the blackboard, how this theorem works in more generality...)

J = J0(43)
D = J.decomposition()
D

[ Simple abelian subvariety 43a(1,43) of dimension 1 of J0(43), Simple abelian subvariety 43b(1,43) of dimension 2 of J0(43) ]

E = D[0]
A = D[1]

A

Simple abelian subvariety 43b(1,43) of dimension 2 of J0(43)

A.modular_kernel()

Finite subgroup with invariants [2, 2] over QQ of Simple abelian subvariety 43b(1,43) of dimension 2 of J0(43)

A.intersection(E)

(Finite subgroup with invariants [2, 2] over QQ of Simple abelian subvariety 43b(1,43) of dimension 2 of J0(43), Simple abelian subvariety of dimension 0 of J0(43))

Example: $J_1(N)$ instead of $J_0(N)$

J = J1(43)
J

Abelian variety J1(43) of dimension 57

%time D = J.decomposition()
# I gave up after 45s; disappointing... but this is why one has to really understand what is going on!

Error in lines 1-1 Traceback (most recent call last): File "/projects/sage/sage-6.10/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 905, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> File "/projects/sage/sage-6.10/local/lib/python2.7/site-packages/sage/modular/abvar/abvar_ambient_jacobian.py", line 367, in decomposition factors = simple_factorization_of_modsym_space(M, simple=simple) File "/projects/sage/sage-6.10/local/lib/python2.7/site-packages/sage/modular/abvar/abvar.py", line 4650, in simple_factorization_of_modsym_space for G in factor_modsym_space_new_factors(M): File "/projects/sage/sage-6.10/local/lib/python2.7/site-packages/sage/modular/abvar/abvar.py", line 4616, in factor_modsym_space_new_factors return [factor_new_space(A) for A in N] File "/projects/sage/sage-6.10/local/lib/python2.7/site-packages/sage/modular/abvar/abvar.py", line 4584, in factor_new_space return t.decomposition() File "/projects/sage/sage-6.10/local/lib/python2.7/site-packages/sage/modular/hecke/hecke_operator.py", line 310, in decomposition D = self.hecke_module_morphism().decomposition() File "/projects/sage/sage-6.10/local/lib/python2.7/site-packages/sage/modules/matrix_morphism.py", line 632, in decomposition E = self.matrix().decomposition(*args,**kwds) File "sage/matrix/matrix_rational_dense.pyx", line 1856, in sage.matrix.matrix_rational_dense.Matrix_rational_dense.decomposition (/projects/sage/sage-6.10/src/build/cythonized/sage/matrix/matrix_rational_dense.c:19616) X = self._decomposition_rational(is_diagonalizable=is_diagonalizable, File "sage/matrix/matrix_rational_dense.pyx", line 1984, in sage.matrix.matrix_rational_dense.Matrix_rational_dense._decomposition_rational (/projects/sage/sage-6.10/src/build/cythonized/sage/matrix/matrix_rational_dense.c:21832) W.echelonize(algorithm = echelon_algorithm, **kwds) File "sage/matrix/matrix_rational_dense.pyx", line 1463, in sage.matrix.matrix_rational_dense.Matrix_rational_dense.echelonize (/projects/sage/sage-6.10/src/build/cythonized/sage/matrix/matrix_rational_dense.c:15830) pivots = self._echelonize_padic() File "sage/matrix/matrix_rational_dense.pyx", line 1613, in sage.matrix.matrix_rational_dense.Matrix_rational_dense._echelonize_padic (/projects/sage/sage-6.10/src/build/cythonized/sage/matrix/matrix_rational_dense.c:17730) pivots, nonpivots, X, d = A._rational_echelon_via_solve() File "sage/matrix/matrix_integer_dense.pyx", line 4485, in sage.matrix.matrix_integer_dense.Matrix_integer_dense._rational_echelon_via_solve (/projects/sage/sage-6.10/src/build/cythonized/sage/matrix/matrix_integer_dense.c:38193) X, d = C._solve_iml(D, right=True) File "sage/matrix/matrix_integer_dense.pyx", line 4179, in sage.matrix.matrix_integer_dense.Matrix_integer_dense._solve_iml (/projects/sage/sage-6.10/src/build/cythonized/sage/matrix/matrix_integer_dense.c:35813) sig_on() File "sage/ext/interrupt/interrupt.pyx", line 88, in sage.ext.interrupt.interrupt.sig_raise_exception (/projects/sage/sage-6.10/src/build/cythonized/sage/ext/interrupt/interrupt.c:924) raise KeyboardInterrupt KeyboardInterrupt

CPU time: 41.55 s, Wall time: 43.47 s

J0(19)

Abelian variety J0(19) of dimension 1

J = J1(19)
J

Abelian variety J1(19) of dimension 7

J.decomposition()

[ Simple abelian subvariety 19aG1(1,19) of dimension 1 of J1(19), Simple abelian subvariety 19bG1(1,19) of dimension 6 of J1(19) ]

︠e5da317b-5940-4e4b-9118-d639209f2200︠

︠825cf85c-06a8-4564-9942-064e6bc04627︠

︠bcc53ca8-ca21-4778-858a-2b15187c7dad︠

︠0f745b1d-4d74-4346-9d58-7e48a317dc01i︠
%md
## My favorite example: $J_0(389)$

My favorite example: $J_0(389)$

J = J0(389); J

Abelian variety J0(389) of dimension 32

%time D = J.decomposition()
D

CPU time: 0.00 s, Wall time: 0.00 s [ Simple abelian subvariety 389a(1,389) of dimension 1 of J0(389), Simple abelian subvariety 389b(1,389) of dimension 2 of J0(389), Simple abelian subvariety 389c(1,389) of dimension 3 of J0(389), Simple abelian subvariety 389d(1,389) of dimension 6 of J0(389), Simple abelian subvariety 389e(1,389) of dimension 20 of J0(389) ]

The smallest-conductor rank 2 elliptic curve if 389a, which is the 1-dimensional factor above.

D[0].intersection(D[1])

(Finite subgroup with invariants [2, 2] over QQ of Simple abelian subvariety 389a(1,389) of dimension 1 of J0(389), Simple abelian subvariety of dimension 0 of J0(389))

D[0].intersection(D[4])

(Finite subgroup with invariants [20, 20] over QQ of Simple abelian subvariety 389a(1,389) of dimension 1 of J0(389), Simple abelian subvariety of dimension 0 of J0(389))

Thus if $A$ is the $20$-dimensional factor above, and we view $E$ and $A$ as abelian subvarieties of $J_0(389)$, then $A \cap E = E[20]$. The intersection, and some Galois cohomology, induces a map $$E(\QQ)/5 E(\QQ) \hookrightarrow \text{Sha}(A/\QQ).$$ This is an example of visibility of Shafarevich-Tate groups, in that $\text{Sha}(A/\QQ)[5]$ is "visible" in terms of points in the Mordell-Weil group of $E$.

E = EllipticCurve('389a'); show(E)
show(E.gens())

$\displaystyle y^2 + y = x^{3} + x^{2} - 2 x$

[$\displaystyle \left(-1 : 1 : 1\right)$, $\displaystyle \left(0 : 0 : 1\right)$]

A = D[4]
L = A.lseries()

L(1)  # heh, why not?

Error in lines 1-1 Traceback (most recent call last): File "/projects/sage/sage-6.10/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 905, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> File "/projects/sage/sage-6.10/local/lib/python2.7/site-packages/sage/modular/abvar/lseries.py", line 95, in __call__ raise NotImplementedError NotImplementedError

L.rational_part() # seriously?

Error in lines 1-1 Traceback (most recent call last): File "/projects/sage/sage-6.10/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 905, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> File "/projects/sage/sage-6.10/local/lib/python2.7/site-packages/sage/modular/abvar/lseries.py", line 152, in rational_part raise NotImplementedError NotImplementedError

12 years ago I started Sage (called Manin then) specifically to re-implement the algorithms from my thesis on an open source foundation. I got distracted along the way, and still haven't finished even that goal, nor has anybody else.

%magma
J := JZero(389);
D := Decomposition(J);
print J, D;

Modular abelian variety JZero(389) of dimension 32 and level 389 over Q [ Modular abelian variety 389A of dimension 1, level 389 and conductor 389 over Q, Modular abelian variety 389B of dimension 2, level 389 and conductor 389^2 over Q, Modular abelian variety 389C of dimension 3, level 389 and conductor 389^3 over Q, Modular abelian variety 389D of dimension 6, level 389 and conductor 389^6 over Q, Modular abelian variety 389E of dimension 20, level 389 and conductor 389^20 over Q ]

%magma
print LRatio(D[5], 1);

51200/97

factor(51200)

2^11 * 5^2