A Refinement of the Faltings-serre Method

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 y + y = x − x. 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 These N are all prime and so this work will indeed superceded by the work of Wiles if his gap can be fixed.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 6= 2. The main result of the first half is Theorem 1 below. Note that there are extensive tables of class numbers and units of cubic fields due to Angell [2] and that information on quartic fields is not required