Cancellation in additively twisted sums on GL(n)

2004/04/29 by Stephen D. Miller, Miller, Stephen D. · 1 citation
Mathematics · #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT) #math.NT #math.RT

paper · pdf · doi:10.48550/arxiv.math/0404521

33 pages; v2 has minor revisions; v3 is 35 pages, has a revised title, and a lengthened introduction

openalex publication_date 2004/04/29 · arxiv created 2004/08/30 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In a previous paper with Schmid (math.NT/0402382) 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 sum_{n < T} a_n exp(2 pi i n alpha) = O_ε(T^3/4+ε) uniformly for alpha real, for a_n the coefficients of the L-function of a cusp form on GL(3,Z)\GL(3,R). We also derive an equivalence (Theorem 7.1) between analogous cancellation statements for cusp forms on GL(n,R), and the sizes of certain period integrals. These in turn imply estimates for the second moment of cusp form L-functions.

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• On Averages of Shifted Convolutions with Applications to GL(2) and GL(3) Fourier Coefficients

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