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An automorphic description of the zeta function of the basic stratum of certain Kottwitz varieties

Yachen Liu

Abstract

We derive formulas for the number of points on the basic stratum of certain Kottwitz varieties in terms of automorphic representations and certain explicit polynomials, for which we present efficient algorithms for computation. We obtain our results using the trace formula, base change, representations of general linear groups over p-adic fields, and a truncation of the formula of Kottwitz for the number of points on Shimura varieties over finite fields.

An automorphic description of the zeta function of the basic stratum of certain Kottwitz varieties

Abstract

We derive formulas for the number of points on the basic stratum of certain Kottwitz varieties in terms of automorphic representations and certain explicit polynomials, for which we present efficient algorithms for computation. We obtain our results using the trace formula, base change, representations of general linear groups over p-adic fields, and a truncation of the formula of Kottwitz for the number of points on Shimura varieties over finite fields.

Paper Structure

This paper contains 28 sections, 53 theorems, 191 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Main results
  4. Proof strategies
  5. Structure of the paper
  6. Local computations
  7. Compact traces
  8. Twisted traces
  9. Some unitary groups
  10. The Kottwitz functions $f_{n\alpha s}, \phi_{n\alpha s}$ and $\varphi_{n \alpha s}$
  11. The functions $(C_{\theta c} \phi_{n \alpha s})^{(P)}$
  12. The functions $\hat{\chi}^{\theta}_{N} \phi_{n \alpha s} ^{(P)}$
  13. Computation of some twisted traces
  14. Twisted traces of $\theta$-stable semi-stable rigid representations
  15. $\theta$-stable semi-stable rigid representations
  16. ...and 13 more sections

Key Result

Theorem 1.2.1

(See Theorem 5.2.1)

Theorems & Definitions (113)

  • Theorem 1.2.1
  • Proposition 1.3.1
  • Remark 1.3.1
  • Proposition 1.3.2
  • Corollary 1.3.1
  • Proposition 1.3.3
  • Proposition 1.3.4
  • Theorem 1.3.1
  • Proposition 1.3.5
  • Definition 2.1.1
  • ...and 103 more