Vanishing of L-functions of Elliptic Curves over Number Fields

Let E be an elliptic curve over Q, with L-function LE(s). For any primitive Dirichlet character χ, let LE(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 LE(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 LE(1, χ) = 0, but that for any fixed prime k ≥ 7, there are only finitely many character of order k such that LE(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.