Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Short sums of trace functions over function fields and their applications

Will Sawin, Mark Shusterman

TL;DR

The paper develops a function-field analogue of Hooley-type cancellation for short sums of trace functions over \\mathbb{F}_q[u], under four structural hypotheses on the trace sheaves (no Artin–Schreier factors, mixed weights, slopes at infinity bounded by 1, and no finitely supported sections). The authors prove a general MainRes bound via a deformation approach and a novel translation-invariance–based vanishing argument for high-degree cohomology, with the Fourier transform playing a key role. This yields concrete results, including a function-field Mordell-type bound on the least-residue problem and a variance bound for short trace sums, and it furnishes a generalized Hooley-type bound for rational-function sums by localizing to prime moduli. The work provides a robust framework for short sums of trace functions in function fields, with potential implications for related integer settings through unconditional or averaged methods, and it highlights the power of étale cohomology and translation-invariance in analytic number-theoretic problems. All findings are presented with explicit bounds and explicit dependence on conductors and ranks, and the techniques bridge cohomological vanishing, Fourier analysis on sheaves, and arithmetic applications.

Abstract

For large enough (but fixed) prime powers $q$, and trace functions to squarefree moduli in $\mathbb{F}_q[u]$ with slopes at most $1$ at infinity, and no Artin--Schreier factors in their geometric global monodromy, we come close to square-root cancellation in short sums. A special case is a function field version of Hooley's Hypothesis $R^*$ for short Kloosterman sums. As a result, we are able to make progress on several problems in analytic number theory over $\mathbb{F}_q[u]$ such as Mordell's problem on the least residue class not represented by a polynomial and the variance of short Kloosterman sums.

Short sums of trace functions over function fields and their applications

TL;DR

The paper develops a function-field analogue of Hooley-type cancellation for short sums of trace functions over \\mathbb{F}_q[u], under four structural hypotheses on the trace sheaves (no Artin–Schreier factors, mixed weights, slopes at infinity bounded by 1, and no finitely supported sections). The authors prove a general MainRes bound via a deformation approach and a novel translation-invariance–based vanishing argument for high-degree cohomology, with the Fourier transform playing a key role. This yields concrete results, including a function-field Mordell-type bound on the least-residue problem and a variance bound for short trace sums, and it furnishes a generalized Hooley-type bound for rational-function sums by localizing to prime moduli. The work provides a robust framework for short sums of trace functions in function fields, with potential implications for related integer settings through unconditional or averaged methods, and it highlights the power of étale cohomology and translation-invariance in analytic number-theoretic problems. All findings are presented with explicit bounds and explicit dependence on conductors and ranks, and the techniques bridge cohomological vanishing, Fourier analysis on sheaves, and arithmetic applications.

Abstract

For large enough (but fixed) prime powers , and trace functions to squarefree moduli in with slopes at most at infinity, and no Artin--Schreier factors in their geometric global monodromy, we come close to square-root cancellation in short sums. A special case is a function field version of Hooley's Hypothesis for short Kloosterman sums. As a result, we are able to make progress on several problems in analytic number theory over such as Mordell's problem on the least residue class not represented by a polynomial and the variance of short Kloosterman sums.
Paper Structure (17 sections, 25 theorems, 184 equations)

This paper contains 17 sections, 25 theorems, 184 equations.

Table of Contents

  1. Introduction
  2. Prior work over $\mathbb{Z}$
  3. Hooley's conjectures on short Kloosterman sums, its variants, and applications
  4. Approaches to incomplete exponential sums and their applications
  5. Additional averaging
  6. More general short exponential sums
  7. Our results over function fields
  8. Short sums of trace functions
  9. Applications
  10. Strategy of the proof of \ref{['MainRes']}
  11. Translation-Invariant Sheaves on $\mathbb A^1$
  12. Vanishing of Cohomology
  13. Proof of \ref{['MainRes']}
  14. Proof of \ref{['GeneralizedHooleyRationalFunction']}
  15. Proof of \ref{['MordellCor']}
  16. ...and 2 more sections

Key Result

Theorem 1.3

Suppose that the sheaves $\{\mathcal{F}_\pi\}_{\pi \mid g}$ satisfy the four assumptions above. Then with the implied constant being $\sqrt q$.

Theorems & Definitions (50)

  • Conjecture 1.1
  • Remark 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Theorem 1.6
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Theorem 2.3
  • ...and 40 more