Bilinear forms with Kloosterman sums and applications

TL;DR

This paper establishes new bounds for bilinear forms involving hyper-Kloosterman sums, with applications to moments of cusp forms and distribution of Eisenstein-Hecke coefficients, using advanced algebraic geometry techniques.

Contribution

It introduces novel bounds for complete sums over finite fields and applies these to problems in analytic number theory involving automorphic forms.

Findings

Derived bounds improve understanding of bilinear forms in hyper-Kloosterman sums.

Applied bounds yield results on moments of cusp forms twisted by characters.

Analyzed distribution of Eisenstein-Hecke coefficients in arithmetic progressions.

Abstract

We prove non-trivial bounds for general bilinear forms in hyper-Kloosterman sums when the sizes of both variables may be below the P\'olya-Vinogradov range. We then derive applications to the second moment of holomorphic cusp forms twisted by characters modulo primes, and to the distribution in arithmetic progressions to large moduli of certain Eisenstein-Hecke coefficients on $\GL_3$. Our main tools are new bounds for certain complete sums in three variables over finite fields, proved using methods from algebraic geometry, especially $\ell$-adic cohomology and the Riemann Hypothesis.