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Symplectic Kloosterman Sums for $\operatorname{Sp}(2n)$ with Powerful Moduli

Gilles Felber

TL;DR

The paper extends symplectic Kloosterman sums to the group $\mathrm{Sp}(2n)$ with powerful moduli, yielding non-trivial bounds when $C$ decomposes into prime-power blocks. It develops a framework combining Smith normal form factorization, a p-adic Taylor expansion, and block decomposition to achieve square-root type cancellation for $K_n(Q,T;C)$. Central technical components include matrix Gauss sums, counting lemmas for quadratic matrix equations, and a careful treatment of the prime 2 case. As an application, it proves effective equidistribution results for coprime symmetric pairs on the Siegel torus, with implications for Petersson-type formulas in higher rank.

Abstract

We prove a non-trivial bound for $\operatorname{Sp}(2n)$ Kloosterman sums of moduli not equal to a prime multiple of the identity. These sums are attached to Siegel modular forms on the group $\operatorname{Sp}(2n)$ and appear in the corresponding Petersson formula. We give an application to equidistribution of coprime symmetric pairs.

Symplectic Kloosterman Sums for $\operatorname{Sp}(2n)$ with Powerful Moduli

TL;DR

The paper extends symplectic Kloosterman sums to the group with powerful moduli, yielding non-trivial bounds when decomposes into prime-power blocks. It develops a framework combining Smith normal form factorization, a p-adic Taylor expansion, and block decomposition to achieve square-root type cancellation for . Central technical components include matrix Gauss sums, counting lemmas for quadratic matrix equations, and a careful treatment of the prime 2 case. As an application, it proves effective equidistribution results for coprime symmetric pairs on the Siegel torus, with implications for Petersson-type formulas in higher rank.

Abstract

We prove a non-trivial bound for Kloosterman sums of moduli not equal to a prime multiple of the identity. These sums are attached to Siegel modular forms on the group and appear in the corresponding Petersson formula. We give an application to equidistribution of coprime symmetric pairs.

Paper Structure

This paper contains 11 sections, 31 theorems, 226 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Acknowledgment
  4. Elementary properties
  5. Symplectic matrices
  6. Factorization of Kloosterman sums
  7. Taylor expansion argument
  8. Block decomposition
  9. Matrix Gauss sums and a counting problem
  10. The case p=2
  11. Application

Key Result

Theorem 1.1

Let $p$ be a prime number and $Q,T$ be two symmetric half-integral matrices. In both cases, the implicit constant only depends on the dimension $n$.

Theorems & Definitions (72)

  • Theorem 1.1
  • Theorem 1.2: MT26TZ25, to appear
  • Theorem 1.3
  • Lemma 2.1
  • proof
  • Remark
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.4
  • proof
  • ...and 62 more