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Algebraicity of \(L\)-values for elliptic curves in a false Tate curve tower. (English) Zbl 1214.11080

Summary: Let \(E\) be an elliptic curve over \(\mathbb Q\), and \(\tau\) an Artin representation over \(\mathbb Q\) that factors through the non-abelian extension \(\mathbb Q(\sqrt[p^n]{m},\mu_{p^n})/\mathbb Q\), where \(p\) is an odd prime and \(n,m\) are positive integers. We show that \(L(E,\tau,1)\), the special value at \(s=1\) of the \(L\)-function of the twist of \(E\) by \(\tau\), divided by the classical transcendental period \(\Omega_{+}^{d^+}|\Omega_{-}^{d^-}|\varepsilon(\tau)\) is algebraic and Galois-equivariant, as predicted by Deligne’s conjecture.

MSC:

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