The $a$-numbers of non-hyperelliptic curves of genus 3 with cyclic automorphism group of order 6
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| Format: | Preprint |
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2021
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| _version_ | 1866914819611820032 |
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| author | Ohashi, Ryo Kudo, Momonari Harashita, Shushi |
| author_facet | Ohashi, Ryo Kudo, Momonari Harashita, Shushi |
| contents | In this paper, we study non-hyperelliptic curves of genus $3$ with cyclic automorphism group of order $6$. Over an algebraically closed field $K$ of characteristic $\neq 2,3$, such curves are written as plane quartics $C_r: x^3 z + y^4 + r y^2 z^2 + z^4 = 0$ with one parameter $r$. As the first main theorem, we show that $r\neq 0,\pm 2$ and give a necessary and sufficient condition with respect to $r$ and $r'$ such that $C_r \cong C_{r'}$. By describing the Hasse-Witt matrix of $C_r$ in terms of a certain Gauss' hypergeometric series, we obtain the second main theorem, where we determine the possible $a$-number of $C_r$, and give the exact number of isomorphism classes over $K$ of such curves attaining the possible maximal $a$-number. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2111_09777 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | The $a$-numbers of non-hyperelliptic curves of genus 3 with cyclic automorphism group of order 6 Ohashi, Ryo Kudo, Momonari Harashita, Shushi Algebraic Geometry Number Theory In this paper, we study non-hyperelliptic curves of genus $3$ with cyclic automorphism group of order $6$. Over an algebraically closed field $K$ of characteristic $\neq 2,3$, such curves are written as plane quartics $C_r: x^3 z + y^4 + r y^2 z^2 + z^4 = 0$ with one parameter $r$. As the first main theorem, we show that $r\neq 0,\pm 2$ and give a necessary and sufficient condition with respect to $r$ and $r'$ such that $C_r \cong C_{r'}$. By describing the Hasse-Witt matrix of $C_r$ in terms of a certain Gauss' hypergeometric series, we obtain the second main theorem, where we determine the possible $a$-number of $C_r$, and give the exact number of isomorphism classes over $K$ of such curves attaining the possible maximal $a$-number. |
| title | The $a$-numbers of non-hyperelliptic curves of genus 3 with cyclic automorphism group of order 6 |
| topic | Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2111.09777 |