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arXiv · 2511.09459

Bilinear forms with trace functions

Étienne Fouvry·Emmanuel Kowalski·Philippe Michel·Will Sawin
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Abstract

We obtain non-trivial bounds for bilinear sums of trace functions below the Pólya-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.

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Étienne Fouvry, Emmanuel Kowalski, Philippe Michel, Will Sawin. 2026-03-11. Bilinear forms with trace functions. https://arxiv.org/abs/2511.09459

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