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Stratification theorems for exponential sums in families

Dante Bonolis, Emmanuel Kowalski, Katharine Woo

TL;DR

The paper surveys stratification theorems for exponential sums over finite fields, focusing on Katz–Laumon and Fouvry–Katz results and their extensions to uniform variants in families. It develops an algebraic-uniformity framework showing how sums can be realized as trace functions of objects on parameter spaces, stratified into open strata where square-root cancellation holds, and it extends these ideas to analytic uniformity via trace-function technology and perverse-sheaf theory. The authors provide a comprehensive strategy (representation, stratification, semiperversity, Betti bounds) and prove algebraic-uniform KL-stratifications for trace functions, with explicit bounds that behave well in families. They then deduce analytic uniform bounds for exponential sums and thin-set problems, including applications to Burgess-type bounds, equidistribution, and integral points on thin sets, and they include an expository appendix to build intuition for multi-variable trace functions. The work advances uniformity results for stratifications in both algebraic and analytic contexts, enabling quantitative control of constants across families and deepens the connection between exponential-sum estimates and the geometry of the parameter spaces.

Abstract

We survey some of the stratification theorems concerning exponential sums over finite fields, especially those due to Katz-Laumon and Fouvry-Katz, as well as some of their applications. Moreover, motivated partly by recent work of Bonolis, Pierce and Woo (arXiv:2505.11226), we prove that these stratification statements admit uniform variants in families, both algebraically and analytically. The paper includes an Appendix by Forey, Fresán and Kowalski (excerpted from arXiv:2109.11961), which provides an elementary intuitive introduction to trace functions in more than one variable over finite fields.

Stratification theorems for exponential sums in families

TL;DR

The paper surveys stratification theorems for exponential sums over finite fields, focusing on Katz–Laumon and Fouvry–Katz results and their extensions to uniform variants in families. It develops an algebraic-uniformity framework showing how sums can be realized as trace functions of objects on parameter spaces, stratified into open strata where square-root cancellation holds, and it extends these ideas to analytic uniformity via trace-function technology and perverse-sheaf theory. The authors provide a comprehensive strategy (representation, stratification, semiperversity, Betti bounds) and prove algebraic-uniform KL-stratifications for trace functions, with explicit bounds that behave well in families. They then deduce analytic uniform bounds for exponential sums and thin-set problems, including applications to Burgess-type bounds, equidistribution, and integral points on thin sets, and they include an expository appendix to build intuition for multi-variable trace functions. The work advances uniformity results for stratifications in both algebraic and analytic contexts, enabling quantitative control of constants across families and deepens the connection between exponential-sum estimates and the geometry of the parameter spaces.

Abstract

We survey some of the stratification theorems concerning exponential sums over finite fields, especially those due to Katz-Laumon and Fouvry-Katz, as well as some of their applications. Moreover, motivated partly by recent work of Bonolis, Pierce and Woo (arXiv:2505.11226), we prove that these stratification statements admit uniform variants in families, both algebraically and analytically. The paper includes an Appendix by Forey, Fresán and Kowalski (excerpted from arXiv:2109.11961), which provides an elementary intuitive introduction to trace functions in more than one variable over finite fields.

Paper Structure

This paper contains 28 sections, 16 theorems, 182 equations.

Table of Contents

  1. Introduction
  2. General context
  3. Stratification statements
  4. Examples of stratification results and applications
  5. Burgess bound
  6. Class number properties
  7. Equidistribution
  8. Integral points on thin sets
  9. Examples of stratifications for explicit families
  10. Linear spaces
  11. Quadratic forms in $n$ variables
  12. The case of smooth forms
  13. A case with a deep stratification
  14. Stratification in families
  15. Algebraic uniformity: general strategy
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $d\geqslant 0$ be the dimension of $V$. There exist non-negative integers $b_i(M)$, defined for $0\leqslant i\leqslant 2d$, and for each $i$, there exist complex numbers $\alpha_{i,j}$ and integers $w_{i,j}$ for $1\leqslant j\leqslant b_i(M)$, such that and $|\alpha_{i,j}|=|k|^{w_{i,j}/2}$. Moreover, for trace functions of the form (eq-char-sum), we have $w_{i,j}\leqslant i$. In fact, more ge

Theorems & Definitions (45)

  • Theorem 1.1: Structure theorem
  • Remark 1.2
  • Theorem 1.3: Fouvry--Katz
  • Remark 1.4
  • Theorem 1.5: Algebraically uniform KL-stratifications
  • Remark 1.6
  • Theorem 1.7: General algebraically uniform KL-stratifications
  • Definition 1.8: General KL-stratification
  • Remark 1.9
  • Definition 1.10: Transverse semiperverse objects
  • ...and 35 more