Bilinear forms with Kloosterman sums, II

E. Kowalski, Ph. Michel, W. Sawin

Published 2018-02-27Version 1

We prove non-trivial bounds for bilinear forms with hyper-Kloosterman sums modulo a prime $q$ which, for both variables of length $M$, are non-trivial as soon as $M\geq q^{3/8+\delta}$ for any $\delta>0$. This range is identical with the best-known results known for simpler monomial exponentials. The proof combines refinements of the analytic tools from our previous paper, and new geometric results. The key geometric idea is a comparison statement that shows that even when the "sum-product" sheaves that appear in the analysis fail to be irreducible, their decomposition reflects that of the "input" sheaves, except for parameters in a high-codimension subset. This property is proved by a subtle interplay between \'etale cohomology in its algebraic and diophantine incarnations.