An Infinite Family of Artin-Schreier Curves with Minimal a-number
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866918013789274112 |
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| author | Shi, Iris Y. |
| author_facet | Shi, Iris Y. |
| contents | Let $p$ be an odd prime and $k$ be an algebraically closed field with characteristic $p$. Booher and Cais showed that the $a$-number of a $\mathbb Z/p \mathbb Z$-Galois cover of curves $ϕ: Y \to X$ must be greater than a lower bound determined by the ramification of $ϕ$. In this paper, we provide evidence that the lower bound is optimal by finding examples of Artin-Schreier curves that have $a$-number equal to its lower bound for all $p$. Furthermore we use formal patching to generate infinite families of Artin-Schreier curves with $a$-number equal to the lower bound in any characteristic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_11201 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An Infinite Family of Artin-Schreier Curves with Minimal a-number Shi, Iris Y. Number Theory Algebraic Geometry Let $p$ be an odd prime and $k$ be an algebraically closed field with characteristic $p$. Booher and Cais showed that the $a$-number of a $\mathbb Z/p \mathbb Z$-Galois cover of curves $ϕ: Y \to X$ must be greater than a lower bound determined by the ramification of $ϕ$. In this paper, we provide evidence that the lower bound is optimal by finding examples of Artin-Schreier curves that have $a$-number equal to its lower bound for all $p$. Furthermore we use formal patching to generate infinite families of Artin-Schreier curves with $a$-number equal to the lower bound in any characteristic. |
| title | An Infinite Family of Artin-Schreier Curves with Minimal a-number |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2411.11201 |