Weil polynomials of small degree
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912480420167680 |
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| author | Marseglia, Stefano |
| author_facet | Marseglia, Stefano |
| contents | Honda and Tate showed that the isogeny classes of abelian varieties of dimension $g$ over a finite field $\mathbb{F}_q$ are classified in terms of $q$-Weil polynomials of degree $2g$, that is, monic integer polynomials whose set of complex roots consists of $g$ conjugate pairs of absolute value $\sqrt{q}$. There are descriptions of the space of such polynomials for $g \leq 5$, but for $g=3$, $4$ and $5$, these results contain mistakes. We correct these statements. Our proofs build on a criterion that determines when a real polynomial has only real roots in terms of the non-necessarily distinct roots of its first derivative. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_24546 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Weil polynomials of small degree Marseglia, Stefano Number Theory 14K15, 12D10, 26C10 Honda and Tate showed that the isogeny classes of abelian varieties of dimension $g$ over a finite field $\mathbb{F}_q$ are classified in terms of $q$-Weil polynomials of degree $2g$, that is, monic integer polynomials whose set of complex roots consists of $g$ conjugate pairs of absolute value $\sqrt{q}$. There are descriptions of the space of such polynomials for $g \leq 5$, but for $g=3$, $4$ and $5$, these results contain mistakes. We correct these statements. Our proofs build on a criterion that determines when a real polynomial has only real roots in terms of the non-necessarily distinct roots of its first derivative. |
| title | Weil polynomials of small degree |
| topic | Number Theory 14K15, 12D10, 26C10 |
| url | https://arxiv.org/abs/2505.24546 |