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