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Bilinear forms with trace functions

Étienne Fouvry, Emmanuel Kowalski, Philippe Michel, Will Sawin

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.

Bilinear forms with trace functions

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.

Paper Structure

This paper contains 34 sections, 39 theorems, 246 equations.

Table of Contents

  1. Introduction
  2. Bilinear sums of trace functions
  3. General statements
  4. An exceptional case
  5. Cubic toroidal moments
  6. Outline of the paper
  7. Algebro-geometric preliminaries
  8. Around Goursat's Lemma
  9. Goursat's Lemma
  10. A criterion for vanishing of coinvariants
  11. The orthogonal case
  12. A property of gallant sheaves
  13. The case of oxozonic sheaves
  14. Diagonal estimates
  15. General estimates for bilinear forms
  16. ...and 19 more sections

Key Result

Theorem 1.1

Assume that $G$ acts irreducibly on $\mathbf{C}^r$ and that one of the following conditions holds: Let $b$, $c$ be non-zero integers. Let $\delta>0$ be a real number and let $1\leqslant M,N\leqslant q/2$ be integers such that There exists $\eta>0$, depending only on $\delta$, such for that for any sequences of complex numbersWe recall that the notation $n\sim N$ means that $N\leqslant n<2N$, and

Theorems & Definitions (84)

  • Theorem 1.1
  • Example 1.1
  • Remark 1.2
  • Remark 1.3
  • Definition 1.2: Gallant groups and sheaves
  • Theorem 1.3: Type I and II estimates
  • Remark 1.4
  • Theorem 1.4
  • Remark 1.5
  • Proposition 1.5
  • ...and 74 more