Gaussian hypergeometric functions and cyclotomic matrices

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 $\ddot{I}‡$ 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 $$B_{q,2}(\ddot{I}{‡}^r):={\left[\ddot{I}{‡}^r(s_i+s_j)+\ddot{I}{‡}^r(s_i-s_j)\right]}_{1\le i,j\le (q-1)/2}.$$ We prove that if $q\equiv 3(\mathrm{mod}4)$, then $$\det (B_{q,2}(χ^r))=\prod_{0\le k\le (q-3)/2}J_q(χ^r,χ^{2k})= \begin{cases} (-1)^{\frac{q-3}{4}}{\bf i}^nG_q(χ^r)^{\frac{q-1}{2}}/\sqrt{q} & \mbox{if}\ r\equiv 1\pmod 2,\\ G_q(χ^r)^{\frac{q-1}{2}}/q & \mbox{if}\ r\equiv 0\pmod 2, \end{cases}$$ where $J_q(\ddot{I}{‡}^r,\ddot{I}{‡}^{2k})$ and $G_q(\ddot{I}{‡}^r)$ are the Jacobi sum and the Gauss sum over ${\mathbb{F}}_q$ respectively.