Abstract. Let E be an elliptic curve over Q with L-function L E (s). We use the random matrix model of Katz and Sarnak to develop a heuristic for the frequency of vanishing of the twisted L-functions L E (1, χ), as χ runs over the Dirichlet characters of order 3 (cubic twists). The heuristic suggests that the number of cubic twists of conductor less than X for which L E (1, χ) vanishes is asymptotic to b E X 1/2 log e E X for some constants b E , e E depending only on E. We also compute explicitely the conjecture of Keating and Snaith about the moments of the special values L E (1, χ) in the family of cubic twists. Finally, we present experimental data which is consistent with the conjectures for the moments and for the vanishing in the family of cubic twists of L E (s).
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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.