Sums of Twisted Gl(2) L-functions over Function Fields

Let K be a function field of odd characteristic, and let π (resp., η) be a cuspidal automorphic representation of GL2(AK ) (resp., GL1(AK )). Then we show that a weighted sum of the twists of L(s, π) by quadratic characters χD , ∑ D L(s, π ⊗ χD) a0(s, π, D) η(D) |D|, is a rational function and has a finite, nonabelian group of functional equations. A similar construction in the noncuspidal case gives a rational function of three variables. We specify the possible denominators and the degrees of the numerators of these rational functions. By rewriting this object as a multiple Dirichlet series, we also give a new description of the weight functions a0(s, π, D) originally considered by D. Bump, S. Friedberg, and J. Hoffstein. 0. Introduction Let π be an automorphic representation on GL2(AK ), where K is a number field. Then it is a remarkable fact that a weighted sum of the L-functions of quadratic twists of π , ∑ a0(s, π, n) L(s, π, χn) |n| , (0.1) is a meromorphic function of two complex variables and satisfies a group of functional equations. (The sum is best described as a sum over the quadratic twists χn attached to divisors n, as formulated by B. Fisher and S. Friedberg in [FF].) Indeed, this was demonstrated by Friedberg and J. Hoffstein [FH] using a Rankin-Selberg construction, following earlier work of Bump, Friedberg, and Hoffstein [BFH1] – [BFH3] giving a different Rankin-Selberg construction. In this paper we present an entirely different way to understand and analyze the series (0.1). We use this method to establish the properties of (0.1) in the function field case, which are rather striking; moreover, the approach applies equally well in the number field case and leads to a substantial simplification of the argument of [FH]. We also study a three-variable analogue of (0.1), where π on GL2(AK ) is replaced by (π1, π2) on GL1(AK )× GL1(AK ). DUKE MATHEMATICAL JOURNAL Vol. 117, No. 3, c © 2003 Received 6 February 2002. 2000 Mathematics Subject Classification. Primary 11F66; Secondary 11F70, 11G20, 11M38. Friedberg’s research supported by National Science Foundation grant number DMS-9970118.