Automorphic Distributions, L-functions, and Voronoi Summation for GL(3)

A ug 2 00 4 Automorphic Distributions , L - functions , and Voronoi Summation for GL ( 3 )

Automorphic distributions , L - functions , and Voronoi summation for GL

for the error term in Gauss’ classical circle problem, improving greatly on Gauss’ own bound O(x1/2). Though Voronoi originally deduced his formulas from Poisson summation in R2, applied to appropriately chosen test functions, one nowadays views his formulas as identities involving the Fourier coefficients of modular forms on GL(2), i.e., modular forms on the complex upper half plane. A discuss...

Voronoi formulas on GL ( n )

In this paper, we give a new, simple, purely analytic proof of the Voronoi formula for Maass forms on GL(3) first derived by Miller and Schmid. Our method is based on two lemmas of the first author and Thillainatesan which appear in their recent non-adelic proof of the converse theorem on GL(3). Using a different, even simpler method we derive Voronoi formulas on GL(n) twisted by additive chara...

2 00 4 Moments of L - functions , periods of cusp forms , and cancellation in additively twisted sums on GL ( n ) Stephen

In a previous paper with Schmid ([29]) we considered the regularity of automorphic distributions for GL(2,R), and its connections to other topics in number theory and analysis. In this paper we turn to the higher rank setting, establishing the nontrivial bound ∑ n≤T an e 2π i nα = Oε(T 3/4+ε), uniformly in α ∈ R, for an the coefficients of the L-function of a cusp form on GL(3,Z)\GL(3,R). We al...

J ul 2 00 4 Moments of L - functions , periods of cusp forms , and cancellation in additively twisted sums on GL ( n )

In a previous paper with Schmid [29] we considered the regularity of automorphic distributions for GL(2,R), and its connections to other topics in number theory and analysis. In this paper we turn to the higher rank setting, establishing the nontrivial bound ∑ n≤T an e 2π i nα = Oε(T 3/4+ε), uniformly in α ∈ R, for an the coefficients of the L-function of a cusp form on GL(3,Z)\GL(3,R). We also...