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

Abstract

We present an algorithm to determine if the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -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, we can associate to an automorphic representation a family of compatible <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l"> <mml:semantics> <mml:mi> ℓ </mml:mi> <mml:annotation encoding="application/x-tex">\ell</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -adic representations. Our algorithm is based on Faltings-Serre’s method to prove that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l"> <mml:semantics> <mml:mi> ℓ </mml:mi> <mml:annotation encoding="application/x-tex">\ell</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -adic Galois representations are isomorphic. Using the algorithm we provide the first examples of modular elliptic curves over imaginary quadratic fields with residual <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -adic image isomorphic to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S 3"> <mml:semantics> <mml:msub> <mml:mi>S</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">S_3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C 3"> <mml:semantics> <mml:msub> <mml:mi>C</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">C_3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .