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On superspecial hyperelliptic curves of genus 5 whose automorphism groups contain $(\mathbb{Z}/2\mathbb{Z})^3$

Ryo Ohashi, Momonari Kudo

Abstract

While the numbers of superspecial curves of genus at most 3 are well understood, and several computational approaches have been developed to count superspecial curves of genus 4 with large automorphism groups, much less is known in higher genera. In this paper, we construct a feasible algorithm to enumerate superspecial hyperelliptic curves of genus 5 whose automorphism groups contain $(\mathbb{Z}/2\mathbb{Z})^3$. We implement and executing our algorithm in Magma, we succeeded in enumerating such superspecial curves in every characteristic $11 < p < 1000$.

On superspecial hyperelliptic curves of genus 5 whose automorphism groups contain $(\mathbb{Z}/2\mathbb{Z})^3$

Abstract

While the numbers of superspecial curves of genus at most 3 are well understood, and several computational approaches have been developed to count superspecial curves of genus 4 with large automorphism groups, much less is known in higher genera. In this paper, we construct a feasible algorithm to enumerate superspecial hyperelliptic curves of genus 5 whose automorphism groups contain . We implement and executing our algorithm in Magma, we succeeded in enumerating such superspecial curves in every characteristic .
Paper Structure (15 sections, 20 theorems, 64 equations, 1 figure, 3 tables, 4 algorithms)

This paper contains 15 sections, 20 theorems, 64 equations, 1 figure, 3 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Classification of hyperelliptic genus-5 curves
  3. General properties of our curves
  4. Detailed analysis for individual curve types
  5. Generic case: ${\rm Aut}(H) \cong \mathbf{C}_2^3$
  6. Special case: ${\rm Aut}(H) \supset \mathbf{C}_2^2 \rtimes \mathbf{C}_4$
  7. Special case: ${\rm Aut}(H) \supset \mathbf{C}_2 \times \mathbf{D}_{12}$
  8. Special case: ${\rm Aut}(H) \supset \mathbf{C}_2 \times \mathbf{A}_4$
  9. Algorithms to enumerate superspecial our curves
  10. Generic case: ${\rm Aut}(H) \cong \mathbf{C}_2^3$
  11. Special case: ${\rm Aut}(H) \supset \mathbf{C}_2^2 \rtimes \mathbf{C}_4$
  12. Special case: ${\rm Aut}(H) \supset \mathbf{C}_2 \times \mathbf{D}_{12}$
  13. Special case: ${\rm Aut}(H) \supset \mathbf{C}_2 \times \mathbf{A}_4$
  14. Experimental results
  15. Concluding remarks

Key Result

Theorem 1.1

There exists a superspecial hyperelliptic curve of genus 5 whose automorphism group contains $(\mathbb{Z}/2\mathbb{Z})^3$ in any characteristic $23 \leq p < 1000$, except for The number of $\accentset{{\cc@style\underline{ }}}{\mathbb{F}}_p$-isomorphism classes of such curves are summarized in Tables tbl:13-450 and tbl:450-1000.

Figures (1)

  • Figure :

Theorems & Definitions (40)

  • Theorem 1.1
  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • proof
  • ...and 30 more