Vanishing of L-functions of elliptic curves over number fields

2004/06/01 by Chantal David, David, Chantal, Jack Fearnley +3 · 1 citation
Mathematics · Physics and Astronomy · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #math-ph #math.MP #math.NT #msc:11G40

paper · pdf · doi:10.48550/arxiv.math/0406012

arxiv created 2004/06/01 · arxiv updated 2009/12/01

Abstract

Let E be an elliptic curve over ℚ, with L-function L_E(s). For any primitive Dirichlet character χ, let L_E(s, χ) be the L-function of E twisted by χ. In this paper, we use random matrix theory to study vanishing of the twisted L-functions L_E(s, χ) at the central value s=1. In particular, random matrix theory predicts that there are infinitely many characters of order 3 and 5 such that L_E(1, χ)=0, but that for any fixed prime k ≥ 7, there are only finitely many character of order k such that L_E(1, χ) vanishes. With the Birch and Swinnerton-Dyer Conjecture, those conjectures can be restated to predict the number of cyclic extensions K/ℚ of prime degree such that E acquires new rank over K.

Citations

• On the frequency of vanishing of quadratic twists of modular L-functions
• Modular Elliptic Curves and Fermat's Last Theorem

Cited by

• p-adic moments of L-functions

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