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Author: William A. Stein
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seminar/nt/20110513

A. Deines: Dembele's Algorithm for Modular Elliptic Curves Over Real Quadratic Fields

May 13, 2011 in Padelford C401 at 3:30pm

Abstract

From the Eichler-Shimura construction we know that for each newform $f$ of weight 2 and level $n$ with rational Fourier coefficients, there exists an elliptic curve $E_f$ over ${\mathbf Q}$ attached to $f$. We can instead work over $F$, a real quadratic number field with narrow class number one, instead of over $\mathbf{Q}$. Let $f$ be a Hilbert newform of weight 2 and level $I$ with rational Fourier coefficients, where $I$ is an integral ideal of $F$. It is a conjecture that for every $f$ there is an elliptic curve $E_f$ over $F$ attached to $f$. Recently Dembele has developed an algorithm which computes the (candidate) elliptic curve $E_f$ under the assumption that the Eichler-Shimura conjecture is true. I will discuss Dembele's algorithm, give an example, and discuss the status of implementing this algorithm in Sage.


2013-05-11 18:34