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An algorithm for modular elliptic curves over real quadratic fields. (English) Zbl 1211.11078

Summary: Let \(F\) be a real quadratic field with narrow class number one, and \(f\) a Hilbert newform of weight 2 and level \(\mathfrak{n}\) with rational Fourier coefficients, where \(\mathfrak{n}\) is an integral ideal of \(F\). By the Eichler–Shimura construction, which is still a conjecture in many cases when \([F:\mathbb Q]>1\), there exists an elliptic curve \(E_f\) over \(F\) attached to \(f\). In this paper, we develop an algorithm that computes the (candidate) elliptic curve \(E_f\) under the assumption that the Eichler–Shimura conjecture is true. We give several illustrative examples that explain among other things how to compute modular elliptic curves with everywhere good reduction. Over real quadratic fields, such curves do not admit any parameterization by Shimura curves, and so the Eichler–Shimura construction is still conjectural in this case.

MSC:

11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11F41 Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
11G05 Elliptic curves over global fields
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