The Gross-Koblitz formula and almost circulant matrices related to Jacobi sums

Abstract

In this paper, we mainly consider arithmetic properties of the cyclotomic matrix $B_p(k)=\left[J_p(\chi^{ki},\chi^{kj})^{-1}\right]_{1\le i,j\le (p-1-k)/k}$, where $p$ is an odd prime, $1\le k<p-1$ is a divisor of $p-1$, $\chi$ is a generator of the group of all multiplicative characters of the finite field $\mathbb{F}_p$ and $J_p(\chi^{ki},\chi^{kj})$ is Jacobi sum over $\mathbb{F}_p$. By using the Gross-Koblitz formula and some $p$-adic tools, we first prove that $$p^{n-2}\det B_p(k)\equiv (-1)^{\frac{(n-1)(p+n-3)}{2}} \left(\frac{1}{k!}\right)^{n-2}\frac{1}{(2k)!}\pmod {p},$$ where $p-1=kn$. By establishing some theories on almost circulant matrices, we show that $$\det B_p(k)=(-1)^{\frac{(n-1)(p+n-1)}{2}}p^{-(n-1)}n^{n-2}a_p(k).$$ Here $a_p(k)$ is the coefficient of $t$ in the minimal polynomial of $\sum_{y\in U_k}(e^{2\pi{\bf i}y/p}-1)$, where $U_k$ is the set of all $k$-th roots of unity over $\mathbb{F}_p$. Also, for $k=1,2$ we obtain explicit expressions of $\det B_p(k)$.