# 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.

## Authors

Emmanuel Kowalski

ETH Zürich - D-MATH, Rämistrasse 101, CH-8092 Zürich, Switzerland

Philippe Michel

EPFL/SB/TAN, Station 8, CH-1015 Lausanne, Switzerland

William Sawin

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