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Matrix Kloosterman sums

Márton Erdélyi, Árpád Tóth

Abstract

We study a family of exponential sums that arises in the study of the horocyclic flow on $\mathrm{GL}_n$. We prove an explicit version of general purity and find optimal bounds for these sums.

Matrix Kloosterman sums

Abstract

We study a family of exponential sums that arises in the study of the horocyclic flow on . We prove an explicit version of general purity and find optimal bounds for these sums.

Paper Structure

This paper contains 29 sections, 58 theorems, 328 equations.

Table of Contents

  1. Introduction
  2. The subject of the paper
  3. Statements of the results
  4. The organization of the paper
  5. Identities via reduction of rank
  6. Splitting to primary components
  7. A parabolic Bruhat decomposition
  8. The sum over the non-trivial cells.
  9. Reduction to $\mathrm{GL}_{n-2}$
  10. Jordan normal forms over $\mathbb{Z}$
  11. The Proof of Theorem \ref{['thm-motivic']}
  12. Estimates
  13. Kloosterman sums over Bruhat cells: reduction to involutions
  14. Finer decomposition of individual Bruhat cells
  15. Some trace calculations
  16. ...and 14 more sections

Key Result

Theorem 1.1

Theorems & Definitions (123)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Remark 2
  • Remark 3
  • ...and 113 more