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On the vanishing of twisted \(L\)-functions of elliptic curves. (English) Zbl 1115.11033
Summary: 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 \(L_E(1, \chi)\), as \(\chi\) 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 \(L_E(1, \chi)\) 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 explicitly the conjecture of Keating and Snaith about the moments of the special values \(L_E(1, \chi)\) 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 \(L_E(s)\).

MSC:
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
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