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<p>In algebraic geometry, superspecial curves are important research objects. While the number of superspecial genus-3 curves in characteristic <i>p</i> is known, the number of hyperelliptic ones among them has not been determined even for small <i>p</i>. In this paper, in order to compute the latter number, we give an explicit algorithm for computing the Richelot isogeny graph of superspecial principally polarized abelian varieties of dimension 3 using theta functions. In particular, one can determine whether a given vertex in the graph corresponds to the Jacobian of a genus-3 curve or not, and restore the defining equation of such a genus-3 curve from its theta constants. Our algorithm enables efficient enumeration of superspecial genus-3 curves, as all operations can be performed in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {F}_{p^2}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <msup> <mi>p</mi> <mn>2</mn> </msup> </msub> </math></EquationSource> </InlineEquation>. By implementing the algorithm in Magma, we successfully counted the number of hyperelliptic curves among them for all primes <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(11 \le p &lt; 100\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>11</mn> <mo>≤</mo> <mi>p</mi> <mo>&lt;</mo> <mn>100</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Listing superspecial curves of genus three using Richelot isogeny graphs

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年8月7日

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  • Ryo Ohashi,
  • Hiroshi Onuki,
  • Momonari Kudo,
  • Ryo Yoshizumi,
  • Koji Nuida

摘要

In algebraic geometry, superspecial curves are important research objects. While the number of superspecial genus-3 curves in characteristic p is known, the number of hyperelliptic ones among them has not been determined even for small p. In this paper, in order to compute the latter number, we give an explicit algorithm for computing the Richelot isogeny graph of superspecial principally polarized abelian varieties of dimension 3 using theta functions. In particular, one can determine whether a given vertex in the graph corresponds to the Jacobian of a genus-3 curve or not, and restore the defining equation of such a genus-3 curve from its theta constants. Our algorithm enables efficient enumeration of superspecial genus-3 curves, as all operations can be performed in \(\mathbb {F}_{p^2}\) F p 2 . By implementing the algorithm in Magma, we successfully counted the number of hyperelliptic curves among them for all primes \(11 \le p < 100\) 11 ≤ p < 100 .