Bouganis, Thanasis; Dokchitser, Vladimir Algebraicity of \(L\)-values for elliptic curves in a false Tate curve tower. (English) Zbl 1214.11080 Math. Proc. Camb. Philos. Soc. 142, No. 2, 193-204 (2007). 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. Cited in 7 Documents MSC: 11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 11G05 Elliptic curves over global fields PDFBibTeX XMLCite \textit{T. Bouganis} and \textit{V. Dokchitser}, Math. Proc. Camb. Philos. Soc. 142, No. 2, 193--204 (2007; Zbl 1214.11080) Full Text: DOI arXiv