Sharedwww / mazur_eisenstein.texOpen in CoCalc
Author: William A. Stein
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580c:14015 14G25 (10D05)
6Mazur, B.
7Modular curves and the Eisenstein ideal.
8Inst. Hautes �tudes Sci. Publ. Math. No. 47 (1977), 33--186 (1978).
9\end{verbatim}
10
11The main subjects treated in this paper are: (i) classification of
12rational points of finite order (respectively, rational isogeny) of
13elliptic curves over a fixed number field $K$, and (ii) study of
14rational points of $X_0(N)$ and its Jacobian $J$, where $X_0(N)$
15denotes the modular curve over $\Q$ associated to $\Gamma_0(N)$.  As
16for the first problem (i), when $K$ is $\Q$, there is a conjecture of
17A. Ogg which asserts that the group of $\Q$-rational points of an
18elliptic curve over $\Q$ is isomorphic to one of the following 15 groups:
19$\Z/m\Z$ ($m\leq 10$ or $m=12$) or $\Z/2\Z\times \Z/2\nu \Z$ ($\nu\leq 4$).
20The author
21verifies this conjecture by showing that there is no elliptic curve
22over $\Q$ which has $\Q$-rational points of order $N$ when $N$
23is prime and $N=11$
24or $N\geq 17$. (Work of D. Kubert had reduced the problem to this case.)
25
26The proof is based on the fact labeled (III) below concerning rational
27points of $J$. To describe the results on the second problem (ii), to
28which most of this paper is devoted, let $N$ be a prime number and let $n$
29be the numerator of $(N - 1)/12$. We hereafter assume that $n > 1$ (or
30equivalently, $N=11$ or $N\geq 17$). It had been proved by Ogg that the
31divisor class of $(0) - (\infty)$ in $J$, where $0$ and $\infty$ denote two
32cusps on $X_0(N)$, has order $n$. One of the main results on the
33structure of $J(\Q)$ is: (I) the torsion part of $J(\Q)$ is a cyclic group
34of order $n$ generated by the class of $(0) - (\infty)$. In addition, the
35following result is obtained: (II) the Shimura subgroup'' is the
36maximal $\bmu$-type'' subgroup of $J$, where $\bmu$-type
37means that it is the
38Cartier 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
40of $X_0(N)$.
41
42To prove these results (both of which had been
44points of $J$, the author introduces the Eisenstein ideal'' in the Hecke
45algebra. Namely, let $\T$ (the Hecke algebra) denote the $\Z$-algebra
46generated by the Hecke operators $T_{\ell}$ ($\ell$ prime, $\neq N$) and the
47involution $w$, acting on the space of cusp forms of weight $2$ with
48respect 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
51on $J$, and one can decompose it (up to $\Q$-isogeny,
52or equivalently $\C$-isogeny by a result of K. Ribet)
53according to the decomposition of $\Spec(\T)$ into its irreducible
54components.
55
56Let $\tilde{J}$ (the Eisenstein quotient of $J$) be the quotient by an
57abelian subvariety of $J$, whose simple factors correspond to the
58irreducible 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
61finite, and the natural map $J\ra \tilde{J}$ induces an isomorphism of the
62torsion part of $J(\Q)$ onto $\tilde{J}(\Q)$. From this, one easily obtains: (IV)
63the group of $\Q$-rational points of $X_0(N)$ is finite (for $N$ as
64above). 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
67free and of positive rank if $\dim J_+ \geq 1$ (the latter assertion
68being in accord with the conjecture of Birch and Swinnerton-Dyer).
69
70To
71obtain these results, one needs a detailed study of the algebra $\T$ and
72the division points of $J$ by ideals of $\T$, especially by $\mathfrak{I}$ and
73the prime ideals containing $\mathfrak{I}$. This is done in Chapter II of
74this paper. The main tools are the theory of (quasi-) finite flat
75group schemes over $\Z$ (Chapter I), and the theory of modular forms over
76rings (the first part of Chapter II). The above results (I) - (V) (and
77others) are then established in Chapter III. Also in the final two
78sections, some relevant results in connection with the earlier works
79of the author are obtained.
80
81For a more detailed survey of the content
82of this paper, the reader is referred to the paper by the author and
83J.-P. Serre (Seminaire Bourbaki (1974/1975), Exp. No. 469, pp. 238 -
84255, 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
87a subsequent paper (Invent. Math. 44 (1978), no. 2, 129 - 162).
88
89                                          Reviewed by M. Ohta
90
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