Bilinear forms with Kloosterman sums and applications

Abstract

We prove nontrivial bounds for general bilinear forms in hyper-Kloosterman sums when the sizes of both variables may be below the range controlled by Fourier-analytic methods (Pólya-Vinogradov range). We then derive applications to the second moment of cusp forms twisted by characters modulo primes, and to the distribution in arithmetic progressions to large moduli of certain Eisenstein-Hecke coefficients on $\mathrm{GL}_3$. 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.

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Authors

Emmanuel Kowalski

ETH Zürich, Departement Mathematik, Zürich, Switzerland, Rämistrasse 101
CH-8092 Zürich, Switzerland

Philippe Michel

Ecole Polytechnique Fédérale de Lausanne, Institute of Mathematics, EPFL, Station 8, CH-1015 Lausanne, Switzerland

Will Sawin

ETH Zürich, Institute for Theoretical Studies, CH-8092 Zürich, Switzerland