Abstract:Let $p$ be an odd prime and $\mathbb{F}_p$ be the finite field with $p$ elements. For any $a,b\in\mathbb{F}_p$, it is known that the Kloosterman sum
$$K_p(a,b)=\sum_{x\in\mathbb{F}_p\setminus\{0\}}e^{\frac{2\pi i}{p}(ax+\frac{b}{x})}$$
can be viewed as a finite field analogue of certain Bessel function. In this paper, using the arithmetic properties of character sums over $\mathbb{F}_p$, we study some cyclotomic matrices involving Kloosterman sums. For example, we prove that the matrix $[K_p(1,i^2+j^2)]_{1\le i,j\le (p-1)/2}$ is singular if and only if $p\ge11$.
From: Hai-Liang Wu [view email]
[v1]
Mon, 1 Jun 2026 06:31:23 UTC (7 KB)
[v2]
Tue, 2 Jun 2026 13:12:51 UTC (7 KB)
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