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Critical values of higher derivatives of twisted elliptic \(L\)-functions. (English) Zbl 1302.11040

Summary: Let \(L(E/\mathbb Q,s)\) be the \(L\)-function of an elliptic curve \(E\) defined over the rational field \(\mathbb Q\). Assuming the Birch–Swinnerton-Dyer conjectures, we examine special values of the \(r\)th derivatives, \(L^{(r)}(E,1,\chi)\), of twists by Dirichlet characters of \(L(E/\mathbb Q,s)\) when \(L(E, 1,\chi)=\ldots=L^{(r-1)}(E,1,\chi)=0\).

MSC:

11G05 Elliptic curves over global fields
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11Y40 Algebraic number theory computations
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