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

TL;DR

This paper introduces an algorithm to verify if the L-series of an automorphic representation matches that of an elliptic curve over an imaginary quadratic field, aiding in proving modularity.

Contribution

It develops a novel algorithm leveraging Faltings-Serre's method to determine the equivalence of Galois representations associated with automorphic forms and elliptic curves over imaginary quadratic fields.

Findings

Algorithm successfully determines L-series equivalence.

Utilizes compatible p-adic representations from automorphic forms.

Provides a practical approach to modularity proofs.

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. \cite{harris-taylor}, \cite{taylorII} and \cite{berger-harcos}) 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.