Bilinear forms with Kloosterman sums and applications

2015/11/30 by E. Kowalski, Ph. Michel, W. Sawin · 1 citation
Mathematics · #math.NT #msc:11T23 #msc:11L05 #msc:11N37 #msc:11N75 #msc:11F66 #msc:14F20 #msc:14D05

paper · pdf

published as Annals of Mathematics, Vol 186, no. 2, September 2017 · v5; 66 pages; minor corrections

arxiv created 2017/03/31 · arxiv updated 2017/04/10

Abstract

We prove non-trivial bounds for general bilinear forms in hyper-Kloosterman sums when the sizes of both variables may be below the Pólya-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 \GL₃. Our main tools are new bounds for certain complete sums in three variables over finite fields, proved using methods from algebraic geometry, especially ℓ-adic cohomology and the Riemann Hypothesis.

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