Bilinear forms with Kloosterman sums and applications

Abstract

We prove nontrivial bounds for general bilinear forms inhyper-Kloosterman sums when the sizes of both variables may be belowthe range controlled by Fourier-analytic methods (Pólya-Vinogradovrange). We then derive applications to the second moment of cuspforms twisted by characters modulo primes, and to the distributionin arithmetic progressions to large moduli of certainEisenstein-Hecke coefficients on $\mathrm{GL}_3$. Our main tools are newbounds for certain complete sums in three variables over finitefields, proved using methods from algebraic geometry, especially$\ell$-adic cohomology and the Riemann Hypothesis.