Cancellation in additively twisted sums on GL(n)

Details Cancellation in additively twisted sums on GL(n) Cancellation in additively twisted sums on GL(n)

2004-04-29

arxiv.org/abs/math/0404521v3

Mathematics

Number Theory

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

Scientific paper

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_\epsilon (T^{3/4+\epsilon}) 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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