Stratification theorems for exponential sums in families

2025/06/23 by Bonolis, Dante, Kowalski, Emmanuel, Woo, Katharine
#11T23 #14F20 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2506.18299

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.

Citations

• Bilinear forms with trace functions
• Counting integral points in thin sets of type II: singularities, sieves, and stratification
• Quantitative sheaf theory
• Burgess bounds for short character sums evaluated at forms II: the mixed case
• Stratification for multiplicative character sums
• On the number of points on a complete intersection over a finite field

Related

• A translation of N.M. Katz's "Exponential Sums"
• A stratification result for an exponential sum modulo p²
• Stratification and averaging for exponential sums: bilinear forms with generalized Kloosterman sums
• On Katz's (A,B)-exponential sums
• Exponential sums over definable subsets of finite fields