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Enumeration Algorithm for Genus-4 Superspecial Hyperelliptic Curves with Automorphism Group $Q_8$

Takara Taniguchi, Ryo Ohashi, Tsuyoshi Takagi

TL;DR

The paper develops explicit necessary and sufficient conditions for isomorphism among genus-4 superspecial hyperelliptic curves with automorphism group containing $Q_8$, and uses these to formulate and implement an enumeration algorithm in Magma that counts isomorphism classes over $\overline{\mathbb{F}}_p$ for primes $7\le p<10^4$. By separating cases according to automorphism type ($C_{16}\rtimes C_2$, $\mathrm{SL}_2(\mathbb{F}_3)$, and $Q_8$), it derives closed-form relations between parameters $a$ and $b$ that determine isomorphism, notably obtaining a complete criterion for the $Q_8$ case: $H_a \cong H_b$ iff $b \in \{\pm a, \pm(2 - \frac{16}{a+2}), \pm(2 + \frac{16}{a-2})\}$. The experimental results up to $p<10^4$ lead to conjectures about the existence and count of such curves, including a predicted $\lfloor p/48\rfloor$ isomorphism classes when $p\equiv 1,7 \pmod{8}$ and nonexistence for $p\equiv 3,5 \pmod{8}$, with further statements about superspecial $\mathrm{SL}_2(\mathbb{F}_3)$ cases and the role of the Cartier-Manin matrix. These findings advance systematic enumeration of genus-4 superspecial curves with prescribed symmetry and pave the way for proofs of the conjectures and broader generalizations.

Abstract

In this paper, we propose an algorithm to enumerate genus-4 superspecial hyperelliptic curves whose automorphism groups isomorphic to the quaternion group. By implementing this algorithm with Magma, we successfully obtain the number of isomorphism classes of such curves in every characteristic $7 \leq p < 10000$. Interestingly, the experimental results lead us to the conjecture that there exist exactly $[p/48]$ isomorphism classes of such curves if $p \equiv 1,7 \pmod{8}$, whereas such curves exist if $p \equiv 3,5 \pmod{8}$

Enumeration Algorithm for Genus-4 Superspecial Hyperelliptic Curves with Automorphism Group $Q_8$

TL;DR

The paper develops explicit necessary and sufficient conditions for isomorphism among genus-4 superspecial hyperelliptic curves with automorphism group containing , and uses these to formulate and implement an enumeration algorithm in Magma that counts isomorphism classes over for primes . By separating cases according to automorphism type (, , and ), it derives closed-form relations between parameters and that determine isomorphism, notably obtaining a complete criterion for the case: iff . The experimental results up to lead to conjectures about the existence and count of such curves, including a predicted isomorphism classes when and nonexistence for , with further statements about superspecial cases and the role of the Cartier-Manin matrix. These findings advance systematic enumeration of genus-4 superspecial curves with prescribed symmetry and pave the way for proofs of the conjectures and broader generalizations.

Abstract

In this paper, we propose an algorithm to enumerate genus-4 superspecial hyperelliptic curves whose automorphism groups isomorphic to the quaternion group. By implementing this algorithm with Magma, we successfully obtain the number of isomorphism classes of such curves in every characteristic . Interestingly, the experimental results lead us to the conjecture that there exist exactly isomorphism classes of such curves if , whereas such curves exist if
Paper Structure (11 sections, 26 theorems, 19 equations, 2 tables, 2 algorithms)

This paper contains 11 sections, 26 theorems, 19 equations, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Analysis
  4. Necessary and sufficient condition
  5. Necessary and sufficient condition when $H_a \cong H_b$ holds
  6. The case where ${\rm Aut} \cong C_{16} \rtimes C_2$
  7. The case where ${\rm Aut} \cong \mathrm{SL}_2(\mathbb{F}_3)$
  8. The case where ${\rm Aut} \cong Q_8$
  9. Enumeration algorithm
  10. Experiment
  11. Conclusion

Key Result

Theorem 1

Let $H$ be genus-$g$ hyperelliptic curve $H:y^2 = f(x)$ over $k$. Then, Cartier-Manin matrix of $H$ is given as where $n := \frac{p-1}{2} \text{deg}\qty(f(x))$, $\qty(f(x))^{\frac{p-1}{2}} = \sum_{i = 0}^{n}c_i x^i$

Theorems & Definitions (34)

  • Definition 1: automorphism
  • Definition 2: superspecial
  • Theorem 1: Cartier-Manin matrix
  • Theorem 2
  • Definition 3: Weierstrass point
  • Theorem 3
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • ...and 24 more