Bilinear forms with trace functions
Number Theory
2026-03-12 v3
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.
Cite
@article{arxiv.2511.09459,
title = {Bilinear forms with trace functions},
author = {Étienne Fouvry and Emmanuel Kowalski and Philippe Michel and Will Sawin},
journal= {arXiv preprint arXiv:2511.09459},
year = {2026}
}
Comments
v3; 60 pages; add result on stability under twists by rank 1 sheaves; some minor changes