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Kloosterman sums on orthogonal groups

Catinca Mujdei

Abstract

We study Kloosterman sums on the orthogonal groups $SO_{3,3}$ and $SO_{4,2}$, associated to short elements of their respective Weyl groups. An explicit description for these sums is obtained in terms of multi-dimensional exponential sums. These are bounded by a combination of methods from algebraic geometry and $p$-adic analysis.

Kloosterman sums on orthogonal groups

Abstract

We study Kloosterman sums on the orthogonal groups and , associated to short elements of their respective Weyl groups. An explicit description for these sums is obtained in terms of multi-dimensional exponential sums. These are bounded by a combination of methods from algebraic geometry and -adic analysis.

Paper Structure

This paper contains 10 sections, 12 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Evaluating Kloosterman sums
  3. $SO_{3,3}$ Kloosterman sums
  4. Plücker coordinates
  5. Explicit description
  6. Bounds
  7. $SO_{4,2}$ Kloosterman sums
  8. Plücker coordinates
  9. Explicit description
  10. Bounds

Key Result

Theorem 1.3

Let $\psi=\psi_{m_1,m_2,m_3}$, $\psi'=\psi_{n_1,n_2,n_3}$ be characters of $U(\mathbb{Q}_p)/U(\mathbb{Z}_p)$, and $r,s\in\mathbb{Z}_{\geq0}$. Then where with $m_1,n_1,n_2\in\mathbb{Z}_p$, $r,s\in\mathbb{Z}_{\geq0}$, $r\leq s$, $x\overline{x}\equiv 1\,(\mathop{\mathrm{mod}}\nolimits p^r)$, $y\overline{y}\equiv 1\,(\mathop{\mathrm{mod}}\nolimits p^{s-r})$.

Theorems & Definitions (18)

  • Example 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4: BFG
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 2.1: S
  • Remark 2.2
  • Proposition 2.3: F
  • Theorem 2.4: F
  • ...and 8 more