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Abstract

Introduction and statement of the main resultLet E be an elliptic curve defined over a number field K. The behavior of the rank of the Lrational points E(L) as L varies over some family of algebraic extensions of K is a problem of fundamental interest. The conjecture of Birch and Swinnerton-Dyer provides a means to investigate this problem via the theory of automorphic L-functions.Assume that the L-function of E coincides with the L-function L(s, π) of a cuspidal automorphic representation of GL (2) of the adèle ring A K . Let L/K be a finite cyclic extension and χ a Galois character of this extension. Then, the conjecture of Birch and Swinnerton-Dyer equates the rank of the χ-isotypic component E(L) χ of E(L) with the order of vanishing of the twisted L-function L(s, π ⊗ χ) at the central point s = 1/2. In particular, the χ-component E(L) χ is finite (according to the conjecture) if and only if the central value L(1/2, π ⊗ χ) is nonzero.Thus, it becomes of arithmetic interest to establish nonvanishing results for central values of twists of automorphic L-functions by characters of finite order. For quadratic twists, this problem has received much attention in recent years. In this paper, we address this question for twists of higher order. Our main result is elaborated in the following theorem.Theorem 1.1. Fix a prime integer n > 2, a number field K containing the nth roots of unity, and a sufficiently large finite set of primes S of K. Let π be a self-contragredient cuspidal automorphic representation of GL(2, A K ) which has trivial central character

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“…The group of functional equations is isomorphic to the dihedral group of order 8. A similar result for higher-order twists may be found in [Brubaker et al 2004].…”

“…In this section we define and prove various properties of the double Dirichlet series. To derive its meromorphic continuation and functional equation we proceed as in [Brubaker et al 2004], but with some simplifications and refinements. We show, for instance, that knowing optimal bounds towards the Ramanujan-Petersson conjecture is not necessary to get optimal regions of convergence.…”

“…Furthermore similar series with higher-order twists instead of the quadratic characters χ n 0 were studied by Brubaker, Bucur, Chinta, Frechette and Hoffstein [Brubaker et al 2004] in order to prove nonvanishing of higher-order twists. To understand the new series Z (s, w, χ , χ ) we follow essentially the program introduced in [Bump et al 1996] to prove the following.…”