Gaussian hypergeometric functions and cyclotomic matrices
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915347267846144 |
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| author | Wu, Hai-Liang Wang, Li-Yuan |
| author_facet | Wu, Hai-Liang Wang, Li-Yuan |
| contents | Let $q=p^n$ be an odd prime power and let $\mathbb{F}_q$ be the finite field with $q$ elements. Let $\widehat{\mathbb{F}_q^{\times}}$ be the group of all multiplicative characters of $\mathbb{F}_q$ and let $χ$ be a generator of $\widehat{\mathbb{F}_q^{\times}}$. In this paper, we investigate arithmetic properties of certain cyclotomic matrices involving nonzero squares over $\mathbb{F}_q$. For example, let $s_1,s_2,\cdots,s_{(q-1)/2}$ be all nonzero squares over $\mathbb{F}_q$. For any integer $1\le r\le q-2$, define the matrix
$$B_{q,2}(χ^r):=\left[χ^r(s_i+s_j)+χ^r(s_i-s_j)\right]_{1\le i,j\le (q-1)/2}.$$
We prove that if $q\equiv 3\pmod 4$, then
$$\det (B_{q,2}(χ^r))=\prod_{0\le k\le (q-3)/2}J_q(χ^r,χ^{2k})=
\begin{cases}
(-1)^{\frac{q-3}{4}}{\bf i}^nG_q(χ^r)^{\frac{q-1}{2}}/\sqrt{q} & \mbox{if}\ r\equiv 1\pmod 2,\\
G_q(χ^r)^{\frac{q-1}{2}}/q & \mbox{if}\ r\equiv 0\pmod 2,
\end{cases}$$
where $J_q(χ^r,χ^{2k})$ and $G_q(χ^r)$ are the Jacobi sum and the Gauss sum over $\mathbb{F}_q$ respectively. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_20583 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Gaussian hypergeometric functions and cyclotomic matrices Wu, Hai-Liang Wang, Li-Yuan Number Theory Let $q=p^n$ be an odd prime power and let $\mathbb{F}_q$ be the finite field with $q$ elements. Let $\widehat{\mathbb{F}_q^{\times}}$ be the group of all multiplicative characters of $\mathbb{F}_q$ and let $χ$ be a generator of $\widehat{\mathbb{F}_q^{\times}}$. In this paper, we investigate arithmetic properties of certain cyclotomic matrices involving nonzero squares over $\mathbb{F}_q$. For example, let $s_1,s_2,\cdots,s_{(q-1)/2}$ be all nonzero squares over $\mathbb{F}_q$. For any integer $1\le r\le q-2$, define the matrix $$B_{q,2}(χ^r):=\left[χ^r(s_i+s_j)+χ^r(s_i-s_j)\right]_{1\le i,j\le (q-1)/2}.$$ We prove that if $q\equiv 3\pmod 4$, then $$\det (B_{q,2}(χ^r))=\prod_{0\le k\le (q-3)/2}J_q(χ^r,χ^{2k})= \begin{cases} (-1)^{\frac{q-3}{4}}{\bf i}^nG_q(χ^r)^{\frac{q-1}{2}}/\sqrt{q} & \mbox{if}\ r\equiv 1\pmod 2,\\ G_q(χ^r)^{\frac{q-1}{2}}/q & \mbox{if}\ r\equiv 0\pmod 2, \end{cases}$$ where $J_q(χ^r,χ^{2k})$ and $G_q(χ^r)$ are the Jacobi sum and the Gauss sum over $\mathbb{F}_q$ respectively. |
| title | Gaussian hypergeometric functions and cyclotomic matrices |
| topic | Number Theory |
| url | https://arxiv.org/abs/2407.20583 |