Experimental Mathematics

On the Vanishing of Twisted L-Functions of Elliptic Curves

Chantal David, Jack Fearnley, and Hershy Kisilevsky

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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 $L_E(s)$}.

Article information

Source
Experiment. Math., Volume 13, Issue 2 (2004), 185-198.

Dates
First available in Project Euclid: 20 July 2004

Permanent link to this document
https://projecteuclid.org/euclid.em/1090350933

Mathematical Reviews number (MathSciNet)
MR2068892

Zentralblatt MATH identifier
1115.11033

Subjects
Primary: 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture [See also 14G10]

Keywords
Elliptic curves L-functions random matrix theory

Citation

David, Chantal; Fearnley, Jack; Kisilevsky, Hershy. On the Vanishing of Twisted L -Functions of Elliptic Curves. Experiment. Math. 13 (2004), no. 2, 185--198. https://projecteuclid.org/euclid.em/1090350933


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