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arXiv · 2606.01759

Cyclotomic matrices related to Kloosterman sums over finite fields

Hai-Liang Wu
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Abstract

In this paper, by using the arithmetic properties of character sums over finite fields, we investigate some cyclotomic matrices involving Kloosterman sums over finite fields. For example, let $$K_q(u)=\sum_{x\in\mathbb{F}_q\setminus\{0\}}e^{\frac{2\pi i}{p}{\rm Tr}_{\mathbb{F}_q/\mathbb{F}_p}\left(x+\frac{u}{x}\right)}$$ be the Kloosterman sum over $\mathbb{F}_q$, where $q=p^f$ is an odd prime power. We prove that matrix $[K_q(s_i+s_j)]_{1\le i,j\le (q-1)/2}$ is singular whenever $q\ge 11$, where $s_1,s_2,\cdots,s_{(q-1)/2}$ are exactly all non-zero squares over $\mathbb{F}_q$.

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Hai-Liang Wu. 2026-06-01. Cyclotomic matrices related to Kloosterman sums over finite fields. https://arxiv.org/abs/2606.01759

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