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