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Matrix Kloosterman sums modulo prime powers

Márton Erdélyi, Árpád Tóth, Gergely Zábrádi

Abstract

We give optimal bounds for matrix Kloosterman sums modulo prime powers extending earlier work of the first two authors on the case of prime moduli. These exponential sums arise in the theory of the horocyclic flow on $\mathrm{GL}_n$.

Matrix Kloosterman sums modulo prime powers

Abstract

We give optimal bounds for matrix Kloosterman sums modulo prime powers extending earlier work of the first two authors on the case of prime moduli. These exponential sums arise in the theory of the horocyclic flow on .
Paper Structure (19 sections, 27 theorems, 93 equations)

This paper contains 19 sections, 27 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Reduction to counting
  4. The case $p^{2l}$
  5. The case $p^{2l+1}$
  6. The regular semisimple case.
  7. Technical background
  8. Partitions
  9. Centralizers in $GL_n(\mathbb{F}_q)$
  10. Sylvester's equation
  11. Generalities on multivariable Gauss sums.
  12. The equation $AX\equiv X^{-1}B$ to prime modulus
  13. Preliminary observations
  14. The proof of Theorem \ref{['thm:quadr-matrix-explicit']} in the invertible cases
  15. The proof of Theorem \ref{['thm:quadr-matrix-explicit']} in the nilpotent case
  16. ...and 4 more sections

Key Result

Proposition 1.1

Assume $\gcd(A,B,p)=1$ and that $k>1$.

Theorems & Definitions (54)

  • Proposition 1.1
  • Remark 1.2
  • Corollary 1.3
  • Proposition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Proposition 1.8
  • Theorem 1.9
  • proof : The proof of Corollary \ref{['cor:Salie-type']}
  • ...and 44 more