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An algorithm for modular elliptic curves over real quadratic fields

Lassina Dembélé

Number 15
Author Lassina Dembélé
Project A09
Year 2008

Let F be a real quadratic field with narrow class number one, and f a Hilbert newform of weight 2 and level n with rational Fourier coefficients, where n is an integal ideal of F. By the Eichler-Shimura construction, which is still a conjecture in many cases when F:Q>1, there exists an elliptic curve Ef over F attached to f. In this paper, we develop an algorithm that computes the elliptic curve Ef. We give several illustrative examples which explain among other things how to compute modular elliptic curves with everywhere good reduction. Such curves do not admit any parametrization by Shimura curves, and so the Eichler-Shimura construction is still conjectural in this case.

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