Introduction

In recent years the classification of elliptic curves over ℚ of various conductors has been attempted. Many results have shown that elliptic curves of a certain conductor do not exist. Later methods have concentrated on small conductors, striving to find them all and hence to verify the Shimura-Taniyama-Weil conjecture for those conductors. A typical case is the conductor 11. In [1], Agrawal, Coates, Hunt, and van der Poorten showed that every elliptic curve over ℚ of conductor 11 is ℚ-isogenous to y2 + y = x3 – x2. Their methods involved a lot of computation and the use of Baker's method. In [12], Serre subsequently applied Faltings' ideas to reprove this result in a much shorter way. He called this approach “the method of quartic fields”.

In this paper I first seek to refine this method and to make it possible to classify elliptic curves over ℚ of conductor N for a large number of N. These N are all prime and so this work is indeed superceded by the result of Wiles that every semistable elliptic curve over ℚ is modular (if fixed). The advantage of my method is that it provides a much simpler approach (when it works). Like Wiles, I am using deformations of Galois representations but in a more elementary way. The second half of the paper indicates how the Faltings-Serre method can be used to describe spaces of Galois representations and gives the first applications of the method to mod p representations with p ≠ 2.