On the Vanishing of Twisted L-Functions of Elliptic Curves

Abstract

Let E be an elliptic curve over q with L-function {\small $L_E(s)$}. We use the random matrix model of Katz and Sarnak to develop a heuristic for the frequency of vanishing of the twisted L-functions {\small $L_E(1, \chi)$}, as {\small $\chi$} 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 {\small $L_E(1, \chi)$} vanishes is asymptotic to {\small $b_E X^{1/2} \log^{e_E}{X}$} for some constants {\small $b_E, e_E$} depending only on E. We also compute explicitly the conjecture of Keating and Snaith about the moments of the special values {\small $L_E(1, \chi)$} 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 {\small $L_E(s)$}.