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 $$p^{n-2}\det B_p(k)\equiv (-1{)}^{\frac{(n-1)(p+n-3)}{2}}{\left(\frac{1}{k!}\right)}^{n-2}\frac{1}{(2k)!}(\mathrm{mod}p),$$ 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 ${\sum }_{y\in U_k}(e^{2\pi \mathbf{i}y/p}-1)$, 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 $\det B_p(k)$.

References in corpus (1)

• On cyclotomic matrices involving Gauss sums over finite fields