Algebraicity of L-values for elliptic curves in a false Tate curve tower

Abstract

AbstractLet E be an elliptic curve over ${\mathbb Q}$, and τ 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,τ,1), the special value at s=1 of the L-function of the twist of E by τ, divided by the classical transcendental period Ω+d+|Ω−d−|ε(τ) is algebraic and Galois-equivariant, as predicted by Deligne's conjecture.