On cyclotomic matrices related to Kloosterman sums over finite fields
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910279880671232 |
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| author | Wu, Hai-Liang |
| author_facet | Wu, Hai-Liang |
| contents | 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
$$K_p(a,b)=\sum_{x\in\mathbb{F}_p\setminus\{0\}}e^{\frac{2πi}{p}(ax+\frac{b}{x})}$$
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 $p\ge11$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_01759 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On cyclotomic matrices related to Kloosterman sums over finite fields Wu, Hai-Liang Number Theory 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 $$K_p(a,b)=\sum_{x\in\mathbb{F}_p\setminus\{0\}}e^{\frac{2πi}{p}(ax+\frac{b}{x})}$$ 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 $p\ge11$. |
| title | On cyclotomic matrices related to Kloosterman sums over finite fields |
| topic | Number Theory |
| url | https://arxiv.org/abs/2606.01759 |