Minimal $a$-numbers of Artin--Schreier covers of ordinary curves
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866911613986013184 |
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| author | Cais, Bryden Ulmer, Douglas |
| author_facet | Cais, Bryden Ulmer, Douglas |
| contents | Let $k$ be a perfect field of characteristic $p>0$, and let $d$ be a positive integer not divisible by $p$. We define a non-empty Zariski open subset $U$ of the space of polynomials of degree $d$, and for $f(x)\in U(k)$, we compute the $a$-number of the curve defined by $y^p-y=f(x)$. This $a$-number realizes a lower bound given by Booher and Cais, so the latter is tight. Our result also implies that the bound of Booher and Cais for minimal $a$-numbers of Artin-Schreier covers of ordinary curves is tight. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_20091 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Minimal $a$-numbers of Artin--Schreier covers of ordinary curves Cais, Bryden Ulmer, Douglas Number Theory Algebraic Geometry 11G20, 14D10, 14H30 Let $k$ be a perfect field of characteristic $p>0$, and let $d$ be a positive integer not divisible by $p$. We define a non-empty Zariski open subset $U$ of the space of polynomials of degree $d$, and for $f(x)\in U(k)$, we compute the $a$-number of the curve defined by $y^p-y=f(x)$. This $a$-number realizes a lower bound given by Booher and Cais, so the latter is tight. Our result also implies that the bound of Booher and Cais for minimal $a$-numbers of Artin-Schreier covers of ordinary curves is tight. |
| title | Minimal $a$-numbers of Artin--Schreier covers of ordinary curves |
| topic | Number Theory Algebraic Geometry 11G20, 14D10, 14H30 |
| url | https://arxiv.org/abs/2604.20091 |