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Cyclotomic matrices and hypergeometric functions over finite fields

Hai-Liang Wu, Yue-Feng She, Li-Yuan Wang

TL;DR

The paper addresses arithmetic properties of determinants of cyclotomic-type matrices over finite fields by leveraging hypergeometric functions over finite fields. It defines the determinant $S_q(r,d)$ for $q=2n+1$ and derives complete, case-by-case evaluations in terms of Gaussian hypergeometric functions ${}_2F_1$ and Jacobi sums, including a vanishing result when $d$ is a non-square with parity constraints. A second main result analyzes the determinant $D(d,q)$ associated to quadratic forms, showing it factors as a square times a field element, with explicit exponent and sign depending on $qmod 4$; in the prime-field setting this yields $D(d,p)=d^{(p+1)/4}(-1)^{(h(-p)-1)/2}z_p(d)^2$, thereby confirming Sun's conjecture. The methods combine permutation signs, eigenvalue decompositions, and identities for Jacobi sums with Greene's finite-field hypergeometric framework, linking determinant arithmetic to analogues of gamma-function products in finite fields.

Abstract

With the help of hypergeometric functions over finite fields, we study some arithmetic properties of cyclotomic matrices involving characters and binary quadratic forms over finite fields. Also, we confirm some related conjectures posed by Zhi-Wei Sun.

Cyclotomic matrices and hypergeometric functions over finite fields

TL;DR

The paper addresses arithmetic properties of determinants of cyclotomic-type matrices over finite fields by leveraging hypergeometric functions over finite fields. It defines the determinant for and derives complete, case-by-case evaluations in terms of Gaussian hypergeometric functions and Jacobi sums, including a vanishing result when is a non-square with parity constraints. A second main result analyzes the determinant associated to quadratic forms, showing it factors as a square times a field element, with explicit exponent and sign depending on ; in the prime-field setting this yields , thereby confirming Sun's conjecture. The methods combine permutation signs, eigenvalue decompositions, and identities for Jacobi sums with Greene's finite-field hypergeometric framework, linking determinant arithmetic to analogues of gamma-function products in finite fields.

Abstract

With the help of hypergeometric functions over finite fields, we study some arithmetic properties of cyclotomic matrices involving characters and binary quadratic forms over finite fields. Also, we confirm some related conjectures posed by Zhi-Wei Sun.

Paper Structure

This paper contains 6 sections, 7 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Notations and Some History on This Topic
  3. Hypergeometric Functions over Finite Fields
  4. Main Results
  5. Proof of Theorem \ref{['Thm. A']}
  6. Proof of Theorem \ref{['Thm. B']}

Key Result

Theorem 1.1

Let $q=2n+1$ be an odd prime power and let $\chi$ be a generator of the cyclic group of all multiplicative characters of $\mathbb{F}_q$. For any $d\in\mathbb{F}_q^{\times}$ and $1\le r\le q-2$, the following results hold. (i) If $d\not\in\mathbb{F}_q^{\times2}$ and $r\equiv n\ ({\rm{mod}}\ 2)$, then (iii) If $d\in\mathbb{F}_q^{\times2}$ and $q\equiv3\ ({\rm{mod}}\ 4)$, then $S_q(r,d)=S_q(r,1)$. Fu

Theorems & Definitions (10)

  • Theorem 1.1
  • Remark 1.1
  • Conjecture 1.1
  • Theorem 1.2
  • Corollary 1.1
  • Lemma 2.1
  • proof
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4