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Bounds for Kloosterman Sums for $\mathrm{GL}_n$

Johannes Linn

TL;DR

This paper addresses the problem of obtaining nontrivial upper bounds for generalized Kloosterman sums on GL_{N+1} across all Weyl elements. It develops a DR98-inspired parametrization with a diagrammatic two-block decomposition and proves an induction on the number of blocks to handle all Weyl elements, yielding a power-saving bound Kl_p(\psi,\psi', frak{n}) \\ll_\\varepsilon C^{l(w)/2} (\\prod_{k=1}^N p^{r_k})^{1-1/(4l(w))+\\varepsilon} (and a variant for \\Gamma_0(q)); a second bound with different character-dependency is also established, namely Kl_p( \psi,\psi',\\mathfrak{n}) \\ll_\\varepsilon C (\\prod p^{r_k})^{1-1/(2N l(w))+\\varepsilon}. The approach combines a precise Bruhat decomposition, a path-diagram encoding of the left/right factors, and multiple applications of the Weil bound to GL_2-like inner sums, with a careful grouping of variables to control interdependencies. The results hold over general non-archimedean local fields and extend to congruence subgroups \\Gamma_0(q). Overall, the work advances the understanding of Kloosterman sums on GL_n by providing explicit, uniformly nontrivial bounds for all Weyl elements, useful for trace formula applications and beyond.

Abstract

In this paper power saving bounds for general Kloosterman sums for all Weyl elements for $\mathrm{GL}_n$ for $n>2$ are proven, improving the trivial bound by Dąbrowski and Reeder. This is achieved by representing the sums in an explicit way as exponential sums and bounding these through applications of the Weil bound.

Bounds for Kloosterman Sums for $\mathrm{GL}_n$

TL;DR

This paper addresses the problem of obtaining nontrivial upper bounds for generalized Kloosterman sums on GL_{N+1} across all Weyl elements. It develops a DR98-inspired parametrization with a diagrammatic two-block decomposition and proves an induction on the number of blocks to handle all Weyl elements, yielding a power-saving bound Kl_p(\psi,\psi', frak{n}) \\ll_\\varepsilon C^{l(w)/2} (\\prod_{k=1}^N p^{r_k})^{1-1/(4l(w))+\\varepsilon} (and a variant for \\Gamma_0(q)); a second bound with different character-dependency is also established, namely Kl_p( \psi,\psi',\\mathfrak{n}) \\ll_\\varepsilon C (\\prod p^{r_k})^{1-1/(2N l(w))+\\varepsilon}. The approach combines a precise Bruhat decomposition, a path-diagram encoding of the left/right factors, and multiple applications of the Weil bound to GL_2-like inner sums, with a careful grouping of variables to control interdependencies. The results hold over general non-archimedean local fields and extend to congruence subgroups \\Gamma_0(q). Overall, the work advances the understanding of Kloosterman sums on GL_n by providing explicit, uniformly nontrivial bounds for all Weyl elements, useful for trace formula applications and beyond.

Abstract

In this paper power saving bounds for general Kloosterman sums for all Weyl elements for for are proven, improving the trivial bound by Dąbrowski and Reeder. This is achieved by representing the sums in an explicit way as exponential sums and bounding these through applications of the Weil bound.

Paper Structure

This paper contains 11 sections, 12 theorems, 192 equations.

Table of Contents

  1. Introduction
  2. Setting
  3. Bruhat decomposition and System of representatives
  4. Structure of the Proof
  5. Weyl elements with only two blocks
  6. General Weyl elements
  7. Bounds
  8. Preparations
  9. Proof of Theorem \ref{['thm:5']}
  10. Character dependency
  11. Congruence Subgroups

Key Result

Theorem 1

Let $\mathfrak{n}=\mathfrak{c}\cdot w\in\mathrm{GL}_{N+1}$ with exponent vector $r$. For an admissible Weyl element $w$ with length $l(w)$, characters $\psi$ and $\psi^\prime$, and all $\varepsilon>0$ the following holds with Also holds for $q$ a power of $p$.

Theorems & Definitions (36)

  • Remark 1
  • Example 1
  • Theorem 1
  • Remark 2
  • Remark 3
  • Theorem 2
  • Remark 4
  • Theorem 3
  • Remark 5
  • Remark 6
  • ...and 26 more