Vanishing of L-functions of elliptic curves over number fields

Abstract

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