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The $a$-numbers of non-hyperelliptic curves of genus 3 with cyclic automorphism group of order 6

Ryo Ohashi, Momonari Kudo, Shushi Harashita Acta Arithmetica MSC: Primary 14H45; Secondary 14H50, 33C05 DOI: 10.4064/aa231014-27-5 Opublikowany online: 17 October 2024

Streszczenie

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 on $r$ and $r’$ for $C_r \cong C_{r’}$. 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 $a$-numbers of $C_r$, and give the exact number of isomorphism classes over $K$ of such curves attaining the possible maximal $a$-number.

Autorzy

  • Ryo OhashiGraduate School of Information Science and Technology
    The University of Tokyo
    113-0033, Tokyo, Japan
    e-mail
  • Momonari KudoDepartment of Information and Communication Engineering
    Faculty of Information Engineering
    Fukuoka Institute of Technology
    811-0295 Fukuoka, Japan
    e-mail
  • Shushi HarashitaGraduate School of Environment and Information Sciences
    Yokohama National University
    240-8501, Yokohama, Japan
    e-mail

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