80c:14015 14G25 (10D05)
Mazur, B.
Modular curves and the Eisenstein ideal.
Inst. Hautes tudes Sci. Publ. Math. No. 47 (1977), 33--186 (1978).

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 Γ_{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/mZ (m ≤ 10 or m = 12) or Z/2Z × Z/2νZ (ν ≤ 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 ≥ 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 ≥ 17). It had been proved by Ogg that the divisor class of (0)−(∞) in J, where 0 and ∞ 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) − (∞). In addition, the following result is obtained: (II) the “Shimura subgroup” is the maximal “μ-type” subgroup of J, where μ-type means that it is the Cartier dual of a constant group, and the Shimura subgroup is a μ -type cyclic subgroup of order n in J obtained from an étale 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_{ℓ} (ℓ prime, ≠ N) and the involution w, acting on the space of cusp forms of weight 2 with respect to Γ_{0}(N). By definition, the Eisenstein ideal ℑ of T is the ideal generated by 1 + ℓ − T_{ℓ} (ℓ prime, ≠ 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 J̃ (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 ℑ. Then it is true that: (III) the Mordell-Weil group J(Q) is finite, and the natural map J → J̃ induces an isomorphism of the torsion part of J(Q) onto J̃(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 → J̃ factors through J → J^{−} , and (V) the Mordell-Weil group J_{+}(Q) is torsion free and of positive rank if dim J_{+} ≥ 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 ℑ and the prime ideals containing 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