\documentclass[12pt]{article}1\include{macros}2\begin{document}3\begin{verbatim}480c:14015 14G25 (10D05)5Mazur, B.6Modular curves and the Eisenstein ideal.7Inst. Hautes �tudes Sci. Publ. Math. No. 47 (1977), 33--186 (1978).8\end{verbatim}910The main subjects treated in this paper are: (i) classification of11rational points of finite order (respectively, rational isogeny) of12elliptic curves over a fixed number field $K$, and (ii) study of13rational points of $X_0(N)$ and its Jacobian $J$, where $X_0(N)$14denotes the modular curve over $\Q$ associated to $\Gamma_0(N)$. As15for the first problem (i), when $K$ is $\Q$, there is a conjecture of16A. Ogg which asserts that the group of $\Q$-rational points of an17elliptic curve over $\Q$ is isomorphic to one of the following 15 groups:18$\Z/m\Z$ ($m\leq 10$ or $m=12$) or $\Z/2\Z\times \Z/2\nu \Z$ ($\nu\leq 4$).19The author20verifies this conjecture by showing that there is no elliptic curve21over $\Q$ which has $\Q$-rational points of order $N$ when $N$22is prime and $N=11$23or $N\geq 17$. (Work of D. Kubert had reduced the problem to this case.)2425The proof is based on the fact labeled (III) below concerning rational26points of $J$. To describe the results on the second problem (ii), to27which most of this paper is devoted, let $N$ be a prime number and let $n$28be the numerator of $(N - 1)/12$. We hereafter assume that $n > 1$ (or29equivalently, $N=11$ or $N\geq 17$). It had been proved by Ogg that the30divisor class of $(0) - (\infty)$ in $J$, where $0$ and $\infty$ denote two31cusps on $X_0(N)$, has order $n$. One of the main results on the32structure of $J(\Q)$ is: (I) the torsion part of $J(\Q)$ is a cyclic group33of order $n$ generated by the class of $(0) - (\infty)$. In addition, the34following result is obtained: (II) the ``Shimura subgroup'' is the35maximal ``$\bmu$-type'' subgroup of $J$, where $\bmu$-type36means that it is the37Cartier dual of a constant group, and the Shimura subgroup is a $\bmu$38-type cyclic subgroup of order $n$ in $J$ obtained from an \'{e}tale covering39of $X_0(N)$.4041To prove these results (both of which had been42conjectured by Ogg) and to obtain more information about the rational43points of $J$, the author introduces the ``Eisenstein ideal'' in the Hecke44algebra. Namely, let $\T$ (the Hecke algebra) denote the $\Z$-algebra45generated by the Hecke operators $T_{\ell}$ ($\ell$ prime, $\neq N$) and the46involution $w$, acting on the space of cusp forms of weight $2$ with47respect to $\Gamma_0(N)$. By definition, the Eisenstein ideal48$\mathfrak I$ of $\T$ is the ideal generated by $1 +\ell - \T_{\ell}$49($\ell$ prime, $\neq N$) and $1+w$. $\T$ naturally acts50on $J$, and one can decompose it (up to $\Q$-isogeny,51or equivalently $\C$-isogeny by a result of K. Ribet)52according to the decomposition of $\Spec(\T)$ into its irreducible53components.5455Let $\tilde{J}$ (the Eisenstein quotient of $J$) be the quotient by an56abelian subvariety of $J$, whose simple factors correspond to the57irreducible components of $\Spec(\T)$ which meet the support of58$\mathfrak{I}$. Then it is true that: (III) the Mordell-Weil group59$J(\Q)$ is60finite, and the natural map $J\ra \tilde{J}$ induces an isomorphism of the61torsion part of $J(\Q)$ onto $\tilde{J}(\Q)$. From this, one easily obtains: (IV)62the group of $\Q$-rational points of $X_0(N)$ is finite (for $N$ as63above). Next, let $J_{+}=(1+w)J$, and let $J^-$ be the quotient of64$J$ by $J_+$. Then it is proved that $J\ra \tilde{J}$ factors through65$J\ra J^-$ , and (V) the Mordell-Weil group $J_+(\Q)$ is torsion66free and of positive rank if $\dim J_+ \geq 1$ (the latter assertion67being in accord with the conjecture of Birch and Swinnerton-Dyer).6869To70obtain these results, one needs a detailed study of the algebra $\T$ and71the division points of $J$ by ideals of $\T$, especially by $\mathfrak{I}$ and72the prime ideals containing $\mathfrak{I}$. This is done in Chapter II of73this paper. The main tools are the theory of (quasi-) finite flat74group schemes over $\Z$ (Chapter I), and the theory of modular forms over75rings (the first part of Chapter II). The above results (I) - (V) (and76others) are then established in Chapter III. Also in the final two77sections, some relevant results in connection with the earlier works78of the author are obtained.7980For a more detailed survey of the content81of this paper, the reader is referred to the paper by the author and82J.-P. Serre (Seminaire Bourbaki (1974/1975), Exp. No. 469, pp. 238 -83255, Lecture Notes in Math., Vol. 514, Springer, Berlin, 1976; MR 5884\#5681). We note finally that the problem (ii) concerning the85$\Q$-rational isogeny (of prime degree) has been solved by the author in86a subsequent paper (Invent. Math. 44 (1978), no. 2, 129 - 162).8788Reviewed by M. Ohta8990\end{document}91