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A p-adic analog of Hasse-Davenport product relation involving epsilon-factors

Dani Szpruch

Abstract

In this paper we prove some generalizations of the classical Hasse-Davenport product relation for certain arithmetic factors defined on p-adic fields, among them one finds the epsilon-factors appearing in Tate's thesis. We then show that these generalizations are equivalent to some representation theoretic identities relating the determinant of ramified local coefficients matrices defined for coverings of SL(2,F) to Plancherel measures and gamma-factors.

A p-adic analog of Hasse-Davenport product relation involving epsilon-factors

Abstract

In this paper we prove some generalizations of the classical Hasse-Davenport product relation for certain arithmetic factors defined on p-adic fields, among them one finds the epsilon-factors appearing in Tate's thesis. We then show that these generalizations are equivalent to some representation theoretic identities relating the determinant of ramified local coefficients matrices defined for coverings of SL(2,F) to Plancherel measures and gamma-factors.
Paper Structure (18 sections, 27 theorems, 116 equations)

This paper contains 18 sections, 27 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and preparations
  3. General notation
  4. Gauss sums
  5. The classical Hasse-Davenport product relation
  6. p-adic fields
  7. $\tau$ and its basic properties
  8. Tate $\gamma$ and $\epsilon$-factors.
  9. Weil index and the metaplectic $\widetilde{\gamma}$-factor
  10. Relations between $\psi$ and some restrictions of $\chi$
  11. Computations of $\tau(\chi,\psi)$ in the case where $e(\chi)\geq 2$
  12. Main results
  13. A representation theoretic application
  14. Groups and representations
  15. Plancherel measures
  16. ...and 3 more sections

Key Result

Lemma 1

If the characteristic of $\mathbb K$ is odd then

Theorems & Definitions (49)

  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Corollary 1
  • proof
  • Proposition 1
  • ...and 39 more