, Volume 327, Issue 2, pp 315-338

Double Dirichlet series and the n-th order twists of Hecke L-series

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Abstract.

Let n≥3 be a fixed integer and let F be a global field containing the n-th roots of unity. In this paper we study the collective behavior of the n-th order twists of a fixed Hecke L-series for F. To do so, we introduce a double Dirichlet series in two complex variables (s, w) which is a weighted sum of the twists, and obtain its meromorphic continuation. We also study related sums of n-th order Gauss sums. These objects together satisfy a nonabelian group of functional equations in (s, w) of order 32.

Mathematics Subject Classification (1991): 11R42, 11F66, 11F70, 11M41, 11R47.
Research supported in part by NSF grants DMS-9970118 (Friedberg), DMS-0088921 (Hoffstein), and DMS-9700542 (Lieman), and by NSA grant MDA 904-03-1-0012 (Friedberg).