Proving modularity for a given elliptic curve over an imaginary quadratic field

Abstract

We present an algorithm to determine if the L-series associated to an automorphic representation and the one associated to an elliptic curve over an imaginary quadratic field agree. By the work of Harris-Soudry-Taylor, Taylor and Berger-Harcos (cf. (HST93), (Tay94) and (BH)) we can associate to an automorphic representation a family of compatible p-adic representations. Our algorithm is based on Faltings-Serre's method to prove that p-adic Galois representations are isomorphic. Modularity for rational elliptic curves was one of the biggest achievements of last century. Little is known for general number fields. In the case of totally real number fields some techniques do apply, but the result is far from being proven. The case of not totally real fields is more intractable to Taylor-Wiles machinery. In this paper we present an algorithm to determine if the L-series associated to an automorphic representation and the one associated to an elliptic curve over an imaginary quadratic field agree or not. The algorithm is based on Faltings-Serre's method to prove isomorphism of p-adic Galois representation. By the work of Harris-Soudry-Taylor, Taylor and Berger-Harcos (cf. (HST93), (Tay94) and (BH)) we can associate to an automorphic representation a family of compatible p-adic representations, and an elliptic curve has such a family of representations as well in the natural way. The paper is organized as follows: on the first section we present the algorithms (which depend on the residual representations). On the second section we review the results of p-adic representations attached to automorphic forms on imaginary quadratic fields. On the third section we explain Falting-Serre's method on Galois representations. On the fourth section we prove that the algorithm gives the right answer. At last we show some examples and some GP code writen for the examples. 2. Algorithm