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

TL;DR

This paper explores the arithmetic properties of cyclotomic matrices related to Jacobi sums using the Gross-Koblitz formula and $p$-adic tools, providing new determinant formulas and explicit cases for certain parameters.

Contribution

It introduces novel formulas for the determinants of cyclotomic matrices involving Jacobi sums, utilizing $p$-adic analysis and almost circulant matrix theory, with explicit results for specific cases.

Findings

Derived congruence relations for determinants modulo p.

Expressed determinants in terms of coefficients of minimal polynomials.

Provided explicit formulas for determinants when k=1,2.

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 $\det B_p(k)$.