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2008 An Algorithm for Modular Elliptic Curves over Real Quadratic Fields
Lassina Dembélé
Experiment. Math. 17(4): 427-438 (2008).


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 $[F:\Q]>1$, 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.


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Lassina Dembélé. "An Algorithm for Modular Elliptic Curves over Real Quadratic Fields." Experiment. Math. 17 (4) 427 - 438, 2008.


Published: 2008
First available in Project Euclid: 27 May 2009

zbMATH: 1211.11078
MathSciNet: MR2484426

Primary: 11-xx
Secondary: 11GXX

Keywords: Elliptic curves , elliptic curves with everywhere good reduction , Hilbert modular forms , Oda conjecture

Rights: Copyright © 2008 A K Peters, Ltd.


Vol.17 • No. 4 • 2008
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