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

Abstract

Let $q$ be an odd prime power and ${\chi }_q$ 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 $$R_q({\chi }_q)=\prod _{0<k<(q-1)/4}\left(J_q({\varphi }_q,{\chi }_q^k)+J_q({\varphi }_q,{\chi }_q^{-k})\right),$$ which is related to the real parts of Jacobi sums. Also, we reveal the connection between $R_q$ and the cyclotomic matrix $${\left[{\varphi }_q(s_i+s_j)\right]}_{1\le i,j\le (q-1)/2},$$ where ${\varphi }_q$ 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 ${\mathbb{F}}_q$.

arXiv comment22 pages. Comments are welcome