Sharedwww / mazur_eisenstein.texOpen in CoCalc
Author: William A. Stein
1
\documentclass[12pt]{article}
2
\include{macros}
3
\begin{document}
4
\begin{verbatim}
5
80c:14015 14G25 (10D05)
6
Mazur, B.
7
Modular curves and the Eisenstein ideal.
8
Inst. Hautes �tudes Sci. Publ. Math. No. 47 (1977), 33--186 (1978).
9
\end{verbatim}
10
11
The main subjects treated in this paper are: (i) classification of
12
rational points of finite order (respectively, rational isogeny) of
13
elliptic curves over a fixed number field $K$, and (ii) study of
14
rational points of $X_0(N)$ and its Jacobian $J$, where $X_0(N)$
15
denotes the modular curve over $\Q$ associated to $\Gamma_0(N)$. As
16
for the first problem (i), when $K$ is $\Q$, there is a conjecture of
17
A. Ogg which asserts that the group of $\Q$-rational points of an
18
elliptic curve over $\Q$ is isomorphic to one of the following 15 groups:
19
$\Z/m\Z$ ($m\leq 10$ or $m=12$) or $\Z/2\Z\times \Z/2\nu \Z$ ($\nu\leq 4$).
20
The author
21
verifies this conjecture by showing that there is no elliptic curve
22
over $\Q$ which has $\Q$-rational points of order $N$ when $N$
23
is prime and $N=11$
24
or $N\geq 17$. (Work of D. Kubert had reduced the problem to this case.)
25
26
The proof is based on the fact labeled (III) below concerning rational
27
points of $J$. To describe the results on the second problem (ii), to
28
which most of this paper is devoted, let $N$ be a prime number and let $n$
29
be the numerator of $(N - 1)/12$. We hereafter assume that $n > 1$ (or
30
equivalently, $N=11$ or $N\geq 17$). It had been proved by Ogg that the
31
divisor class of $(0) - (\infty)$ in $J$, where $0$ and $\infty$ denote two
32
cusps on $X_0(N)$, has order $n$. One of the main results on the
33
structure of $J(\Q)$ is: (I) the torsion part of $J(\Q)$ is a cyclic group
34
of order $n$ generated by the class of $(0) - (\infty)$. In addition, the
35
following result is obtained: (II) the ``Shimura subgroup'' is the
36
maximal ``$\bmu$-type'' subgroup of $J$, where $\bmu$-type
37
means that it is the
38
Cartier dual of a constant group, and the Shimura subgroup is a $\bmu$
39
-type cyclic subgroup of order $n$ in $J$ obtained from an \'{e}tale covering
40
of $X_0(N)$.
41
42
To prove these results (both of which had been
43
conjectured by Ogg) and to obtain more information about the rational
44
points of $J$, the author introduces the ``Eisenstein ideal'' in the Hecke
45
algebra. Namely, let $\T$ (the Hecke algebra) denote the $\Z$-algebra
46
generated by the Hecke operators $T_{\ell}$ ($\ell$ prime, $\neq N$) and the
47
involution $w$, acting on the space of cusp forms of weight $2$ with
48
respect to $\Gamma_0(N)$. By definition, the Eisenstein ideal
49
$\mathfrak I$ of $\T$ is the ideal generated by $1 +\ell - \T_{\ell}$
50
($\ell$ prime, $\neq N$) and $1+w$. $\T$ naturally acts
51
on $J$, and one can decompose it (up to $\Q$-isogeny,
52
or equivalently $\C$-isogeny by a result of K. Ribet)
53
according to the decomposition of $\Spec(\T)$ into its irreducible
54
components.
55
56
Let $\tilde{J}$ (the Eisenstein quotient of $J$) be the quotient by an
57
abelian subvariety of $J$, whose simple factors correspond to the
58
irreducible components of $\Spec(\T)$ which meet the support of
59
$\mathfrak{I}$. Then it is true that: (III) the Mordell-Weil group
60
$J(\Q)$ is
61
finite, and the natural map $J\ra \tilde{J}$ induces an isomorphism of the
62
torsion part of $J(\Q)$ onto $\tilde{J}(\Q)$. From this, one easily obtains: (IV)
63
the group of $\Q$-rational points of $X_0(N)$ is finite (for $N$ as
64
above). Next, let $J_{+}=(1+w)J$, and let $J^-$ be the quotient of
65
$J$ by $J_+$. Then it is proved that $J\ra \tilde{J}$ factors through
66
$J\ra J^-$ , and (V) the Mordell-Weil group $J_+(\Q)$ is torsion
67
free and of positive rank if $\dim J_+ \geq 1$ (the latter assertion
68
being in accord with the conjecture of Birch and Swinnerton-Dyer).
69
70
To
71
obtain these results, one needs a detailed study of the algebra $\T$ and
72
the division points of $J$ by ideals of $\T$, especially by $\mathfrak{I}$ and
73
the prime ideals containing $\mathfrak{I}$. This is done in Chapter II of
74
this paper. The main tools are the theory of (quasi-) finite flat
75
group schemes over $\Z$ (Chapter I), and the theory of modular forms over
76
rings (the first part of Chapter II). The above results (I) - (V) (and
77
others) are then established in Chapter III. Also in the final two
78
sections, some relevant results in connection with the earlier works
79
of the author are obtained.
80
81
For a more detailed survey of the content
82
of this paper, the reader is referred to the paper by the author and
83
J.-P. Serre (Seminaire Bourbaki (1974/1975), Exp. No. 469, pp. 238 -
84
255, Lecture Notes in Math., Vol. 514, Springer, Berlin, 1976; MR 58
85
\#5681). We note finally that the problem (ii) concerning the
86
$\Q$-rational isogeny (of prime degree) has been solved by the author in
87
a subsequent paper (Invent. Math. 44 (1978), no. 2, 129 - 162).
88
89
Reviewed by M. Ohta
90
91
\end{document}