In the days before [W, TW, BCDT], how did people show that specific elliptic curves over $\mathbb{Q}$ were modular? For instance, I was reading through a paper of Buhler, Gross and Zagier from 1985 on the curve 5077a, and they say that modularity can be checked by a finite computation in the 422-dimensional space of cuspforms of weight 2 and level 5077 (and remark at the end that Serre and Mestre have checked it). A google search brought up the name "Faltings-Serre method": was this the technique of choice? Also, are there any good references for it?



How to show modularity of an elliptic curve?
They explicitly computed quotients of $X\_0(N)$ and identified them with elliptic curves.
Suppose you can compute the space $S\_2(\Gamma\_0(N),\mathbf{C})$ of modular forms. An (isogeny class of) elliptic curves of conductor $N$ corresponds (by modularity) to a normalized new Hecke eigenform with coefficients in $\mathbf{Z}$. Given such an $f$, one can compute the periods of $f$, which allows one to write down a Weierstrass equation for $f$. If one can do this in such a way to guarantee that the coefficients of the Weierstrass equation are *integers*, this allows one to computationally determine exactly the modular elliptic curves of conductor $N$. All this is very well explained in Cremona's book, which is available free online:


<http://www.warwick.ac.uk/~masgaj/book/amec.html>


The particular method referred to in the paper of [BGZ] was presumably the "method of graphs", see for example here:


<http://modular.math.washington.edu/msri06/refs/mestre-method-of-graphs/mestre-en.pdf>


The Faltings-Serre method is slightly different; it allows you to determine, given two Galois representations $\rho\_1$ and $\rho\_2$ from
$G\_{K}$ to (say) $\mathrm{GL}\_n(\mathbf{Z}\_p)$ such that:


1. $\overline{\rho}\_1 \simeq \overline{\rho}\_2$,
2. $\rho\_1$ and $\rho\_2$ are both unramified outside some finite (given) set of places $S$ of $K$.


whether $\rho\_1 \simeq \rho\_2$. You could use it to prove that an elliptic curve $E/\mathbf{Q}$ is modular by comparing the Galois representation attached to the $p$-adic Tate module of $E$ and the Galois representation attached to the conjecturally corresponding eigenform of level $N$ (EDIT: this works because the Galois representation determines the isogeny class of $E$ by Faltings' proof of the Tate conjecture for abelian varieties). However, this wouldn't be so efficient. However, there *are* other situations (for example, elliptic curves over other number fields) where the corresponding automophic form is not enough to compute the periods. In this case, the Faltings-Serre method can be used to prove modularity. Taylor uses this to prove that a certain elliptic curve over $\mathbf{Q}(\sqrt{-3})$ is modular.


