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Symmetry of hypergeometric functions over finite fields and geometric interpretation

Akio Nakagawa

Abstract

We begin by defining general hypergeometric functions over finite fields and obtaining a finite field analogue of a classical symmetry in their complex counterparts. We give a geometric proof for the symmetry by constructing isomorphisms between certain algebraic varieties. The numbers of rational points on these varieties are hypergeometric functions over finite fields.

Symmetry of hypergeometric functions over finite fields and geometric interpretation

Abstract

We begin by defining general hypergeometric functions over finite fields and obtaining a finite field analogue of a classical symmetry in their complex counterparts. We give a geometric proof for the symmetry by constructing isomorphisms between certain algebraic varieties. The numbers of rational points on these varieties are hypergeometric functions over finite fields.
Paper Structure (23 sections, 25 theorems, 287 equations, 1 table)

This paper contains 23 sections, 25 theorems, 287 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Recalls for hypergeometric functions over finite fields
  4. Discrete Fourier transform
  5. General hypergeometric functions over finite fields
  6. Definition and properties
  7. Particular cases
  8. $k=2, n=4$
  9. $k=2, n\geq 5$
  10. Hypergeometric varieties
  11. The number of rational points
  12. General hypergeometric functions
  13. One variable hypergeometric functions
  14. Appell-Lauricella functions
  15. Humbert's functions
  16. ...and 8 more sections

Key Result

Theorem A

For $w \in W_\Delta$, Here, $\chi {}^tw$ is a suitable character of $H_\Delta$.

Theorems & Definitions (43)

  • Theorem A
  • Theorem B: Theorems \ref{['N of X_D(z)-1']}, \ref{['isom fw']} and \ref{['N of X_D(z)']} \ref{['N of X_D(z)-4']}
  • Theorem C: Theorems \ref{['N of X_mn']} and \ref{['isom between X22 and X22-sigma']}
  • Theorem D: Theorems \ref{['N of X_mn']} and \ref{['N of X12']}
  • Definition 2.2
  • Proposition 2.5
  • proof
  • Proposition 3.1
  • proof
  • Definition 3.2
  • ...and 33 more