Bilinear forms with trace functions

TL;DR

This paper establishes new bounds for bilinear sums of trace functions in number theory, leveraging geometric monodromy group properties and advanced stratification techniques, extending beyond classical special cases.

Contribution

It introduces a general stratification theorem and a robust Goursat-Kolchin-Ribet criterion to analyze bilinear trace function sums under minimal assumptions.

Findings

Derived non-trivial bounds below the Pólya-Vinogradov range

Extended analysis to general trace functions beyond classical cases

Developed a new stratification approach for sum analysis

Abstract

We obtain non-trivial bounds for bilinear sums of trace functions below the P\'olya-Vinogradov range assuming only that the geometric monodromy group of the underlying ell-adic sheaf satisfies certain simple structural properties, in contrast to previous works which handled only special cases of Kloosterman and hypergeometric sheaves. Our approach builds on a general "soft" stratification theorem for sums of products of trace functions, based on an idea of Junyan Xu, combined with a new robust version of the Goursat-Kolchin-Ribet criterion.