A characterization of Jacobi sums
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909399859068928 |
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| author | Snowden, Andrew |
| author_facet | Snowden, Andrew |
| contents | Let $\mathbb{M}$ be the group of multiplicative characters of a finite field $\mathbb{F}$, and let $\mathbb{J}(α, β)$ be the Jacobi sum, for $α, β\in \mathbb{M}$. We observe that the function $\mathbb{J} \colon \mathbb{M} \times \mathbb{M} \to \mathbf{C}$ satisfies three elementary properties. We show that these properties (very nearly) characterize Jacobi sums: if $M$ is an arbitrary non-trivial finite abelian group and $J \colon M \times M \to \mathbf{C}$ is a function satisfying these properties then $M$ is naturally the group of multiplicative characters of a finite field and $J$ is the Jacobi sum. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_15011 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A characterization of Jacobi sums Snowden, Andrew Number Theory Let $\mathbb{M}$ be the group of multiplicative characters of a finite field $\mathbb{F}$, and let $\mathbb{J}(α, β)$ be the Jacobi sum, for $α, β\in \mathbb{M}$. We observe that the function $\mathbb{J} \colon \mathbb{M} \times \mathbb{M} \to \mathbf{C}$ satisfies three elementary properties. We show that these properties (very nearly) characterize Jacobi sums: if $M$ is an arbitrary non-trivial finite abelian group and $J \colon M \times M \to \mathbf{C}$ is a function satisfying these properties then $M$ is naturally the group of multiplicative characters of a finite field and $J$ is the Jacobi sum. |
| title | A characterization of Jacobi sums |
| topic | Number Theory |
| url | https://arxiv.org/abs/2411.15011 |