Dirichlet twists ofGLn-automorphicL-functions and hyper-Kloosterman Dirichlet series

Jeanine Van Order

doi:10.5802/afst.1687

Abstract

We calculate mean values of $\operatorname{GL}_n$-automorphic $L$-functions twisted by primitive even Dirichlet characters of prime-power conductor, at arbitrary points within the critical strip, by derivation of special Voronoi summation formulae. Our calculation is novel in that the twisted sum can be expressed in terms of the average itself, and also that it sees the derivation of various new summation formulae in the setting of prime-power modulus. One consequence, as we explain, is to show the analytic continuation and additive summation formulae for hyper-Kloosterman Dirichlet series associated to $\operatorname{GL}_n$-automorphic $L$-functions.

Highlights

Let π = ⊗vπv be a cuspidal automorphic representation of GLn(AQ) of conductor N and unitary central character ω for n 2
Note as well that we restrict to the setting of cuspidal representations for simplicity, and that a similar summation formula could be derived for coefficients of Eisenstein series
Let us first consider the sum over primitive characters χ mod pβ, which via the Möbius inversion formula

Summary

Introduction

Such summation formulae are not accessible via any of the existing works on Voronoi, among them those of Miller–Schmidt [11], Goldfeld–Li [3, 2] or Ichino–Templier [4], or the more recent works of Miller– Zhou [12] and Kiral–Zhou [7] This is a consequence of the delicate analysis required to deal with the implicit and non-admissible choice of archimedean weight function, which leads to the (indirect) derivation of the residual term Xβ(π, δ).(1) Unlike these other works, we make use of the setting of prime-power modulus, where the hyper-Kloosterman sums which appear after unraveling the n-th power Gauss sums can be evaluated explicitly in (1) The aforementioned works require smooth and compactly supported test functions, or else work directly on the level of Dirichlet series in the range of absolute convergence. Note as well that we restrict to the setting of cuspidal representations for simplicity, and that a similar summation formula could be derived for coefficients of Eisenstein series In this way, our calculations should imply the analytic continuation and corresponding functional equations for Eisenstein series on GLn(AQ) twisted by additive characters and hyper-Kloosterman sums.

Some background

Functional equations

Average values

Preliminary estimates

Evaluation of hyper-Kloosterman sums

Reduction to twists by additive characters

Voronoi summation for additive twists

Functional identities for additive twists

Derivation of formulae

Some estimates

Some remarks on hyper-Kloosterman Dirichlet series

Hyper-Kloosterman Dirichlet series at large