An Algorithm for Modular Elliptic Curves over Real Quadratic Fields

Abstract

Let F be a real quadratic field with narrow class number one,and ƒ a Hilbert newform of weight 2 and level n with rationalFourier coefficients, where n is an integral ideal of F. By theEichler–Shimura construction,which is still a conjecture in manycases when [F : ℚ] > 1, there exists an elliptic curve E ƒ over Fattached to ƒ. In this paper, we develop an algorithm that computesthe (candidate) elliptic curve E ƒ under the assumption thatthe Eichler–Shimura conjecture is true. We give several illustrativeexamples that explain among other things how to computemodular elliptic curves with everywhere good reduction. Overreal quadratic fields, such curves do not admit any parameterizationby Shimura curves, and so the Eichler–Shimura constructionis still conjectural in this case.