Products involving the real parts of Jacobi sums and related cyclotomic matrices

Abstract

Let $q$ be an odd prime power and $χ_q$ be a generator of the group of all multiplicative characters of $\mathbb{F}_q$. In this paper, we study the arithmetic properties of the product $$R_q(χ_q)=\prod_{0<k<(q-1)/4}\left(J_q(ϕ_q,χ_q^k)+J_q(ϕ_q,χ_q^{-k})\right),$$ which is related to the real parts of Jacobi sums. Also, we reveal the connection between $R_q$ and the cyclotomic matrix $$\left[ϕ_q(s_i+s_j)\right]_{1\le i,j\le (q-1)/2},$$ where $ϕ_q$ is the unique quadratic multiplicative character of $\mathbb{F}_q$, and $s_1,s_2,\cdots,s_{(q-1)/2}$ are exactly all non-zero squares over $\mathbb{F}_q$.