On a Galois cover of the Hermitian curve of genus $\mathfrak{g}=\frac{1}{8}(q-1)^2$
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| Format: | Preprint |
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2024
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| _version_ | 1866916459471437824 |
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| author | Gatti, Barbara Schulte, Gioia |
| author_facet | Gatti, Barbara Schulte, Gioia |
| contents | In the study of algebraic curves with many points over a finite field, a well known general problem is to understanding better the properties of $\mathbb{F}_{q^2}$-maximal curves whose genera fall in the higher part of the spectrum of the genera of all $\mathbb{F}_{q^2}$-maximal curves. This problem is still open for genera smaller than $ \lfloor \frac{1}{6}(q^2-q+4) \rfloor$. In this paper we consider the case of $\mathfrak{g}=\frac{1}{8}(q-1)^2$ where $q\equiv 1\pmod{4}$ and the curve is the Galois cover of the Hermitian curve w.r.t to a cyclic automorphism group of order $4$. Our contributions concern Frobenius embedding, Weierstrass semigroups and automorphism groups. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_22037 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On a Galois cover of the Hermitian curve of genus $\mathfrak{g}=\frac{1}{8}(q-1)^2$ Gatti, Barbara Schulte, Gioia Algebraic Geometry In the study of algebraic curves with many points over a finite field, a well known general problem is to understanding better the properties of $\mathbb{F}_{q^2}$-maximal curves whose genera fall in the higher part of the spectrum of the genera of all $\mathbb{F}_{q^2}$-maximal curves. This problem is still open for genera smaller than $ \lfloor \frac{1}{6}(q^2-q+4) \rfloor$. In this paper we consider the case of $\mathfrak{g}=\frac{1}{8}(q-1)^2$ where $q\equiv 1\pmod{4}$ and the curve is the Galois cover of the Hermitian curve w.r.t to a cyclic automorphism group of order $4$. Our contributions concern Frobenius embedding, Weierstrass semigroups and automorphism groups. |
| title | On a Galois cover of the Hermitian curve of genus $\mathfrak{g}=\frac{1}{8}(q-1)^2$ |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2410.22037 |