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2004 On the Vanishing of Twisted L-Functions of Elliptic Curves
Chantal David, Jack Fearnley, Hershy Kisilevsky
Experiment. Math. 13(2): 185-198 (2004).


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 $L_E(s)$}.


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Chantal David. Jack Fearnley. Hershy Kisilevsky. "On the Vanishing of Twisted L-Functions of Elliptic Curves." Experiment. Math. 13 (2) 185 - 198, 2004.


Published: 2004
First available in Project Euclid: 20 July 2004

zbMATH: 1115.11033
MathSciNet: MR2068892

Primary: 11G40

Keywords: Elliptic curves , L-functions , Random matrix theory

Rights: Copyright © 2004 A K Peters, Ltd.


Vol.13 • No. 2 • 2004
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