Sharedwww / mazur_eisenstein.texOpen in CoCalc
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 $\Gamma_0(N)$.  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

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). 

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