Gaussian hypergeometric functions and cyclotomic matrices

TL;DR

This paper explores the arithmetic properties of cyclotomic matrices over finite fields, deriving explicit formulas for their determinants involving Gauss and Jacobi sums, especially when the field size satisfies certain congruences.

Contribution

It provides new explicit determinant formulas for cyclotomic matrices over finite fields using Gauss and Jacobi sums, extending understanding of their arithmetic properties.

Findings

Determinant formulas depend on the parity of r and the congruence class of q mod 4.

Explicit expressions involve Gauss sums G_q and Jacobi sums J_q.

Results apply to matrices constructed from nonzero squares over finite fields.

Abstract

Let $q=p^n$ be an odd prime power and let $\mathbb{F}_q$ be the finite field with $q$ elements. Let $\widehat{\mathbb{F}_q^{\times}}$ be the group of all multiplicative characters of $\mathbb{F}_q$ and let $\chi$ be a generator of $\widehat{\mathbb{F}_q^{\times}}$. In this paper, we investigate arithmetic properties of certain cyclotomic matrices involving nonzero squares over $\mathbb{F}_q$. For example, let $s_1,s_2,\cdots,s_{(q-1)/2}$ be all nonzero squares over $\mathbb{F}_q$. For any integer $1\le r\le q-2$, define the matrix $$B_{q,2}(\chi^r):=\left[\chi^r(s_i+s_j)+\chi^r(s_i-s_j)\right]_{1\le i,j\le (q-1)/2}.$$ We prove that if $q\equiv 3\pmod 4$, then $$\det (B_{q,2}(\chi^r))=\prod_{0\le k\le (q-3)/2}J_q(\chi^r,\chi^{2k})= \begin{cases} (-1)^{\frac{q-3}{4}}{\bf i}^nG_q(\chi^r)^{\frac{q-1}{2}}/\sqrt{q} & \mbox{if}\ r\equiv 1\pmod 2,\\ G_q(\chi^r)^{\frac{q-1}{2}}/q & \mbox{if}\ r\equiv 0\pmod 2, \end{cases}$$ where $J_q(\chi^r,\chi^{2k})$ and $G_q(\chi^r)$ are the Jacobi sum and the Gauss sum over $\mathbb{F}_q$ respectively.