Efficient search for superspecial hyperelliptic curves of genus four with automorphism group containing $\mathbb{Z}_6$

TL;DR

This paper introduces an efficient algorithm to find superspecial hyperelliptic curves of genus four with specific automorphism groups, successfully enumerating such curves for primes up to 1000 and classifying their automorphism groups.

Contribution

It presents a new algorithm with improved complexity for enumerating superspecial hyperelliptic curves of genus four with automorphism group containing $bZ_6$, and provides a classification of these curves based on automorphism groups.

Findings

Enumerated superspecial hyperelliptic curves for primes up to 1000.

Found a superspecial hyperelliptic curve for every prime p ≡ 2 mod 3.

Classified hyperelliptic curves of genus 4 by automorphism groups.

Abstract

In arithmetic and algebraic geometry, superspecial (s.sp.\ for short) curves are one of the most important objects to be studied, with applications to cryptography and coding theory. If $g \geq 4$, it is not even known whether there exists such a curve of genus $g$ in general characteristic $p > 0$, and in the case of $g=4$, several computational approaches to search for those curves have been proposed. In the genus-$4$ hyperelliptic case, Kudo-Harashita proposed a generic algorithm to enumerate all s.sp.\ curves, and recently Ohashi-Kudo-Harashita presented an algorithm specific to the case where automorphism group contains the Klein 4-group. In this paper, we propose an algorithm with complexity $\tilde{O}(p^4)$ in theory but $\tilde{O}(p^3)$ in practice to enumerate s.sp.\ hyperelliptic curves of genus 4 with automorphism group containing the cyclic group of order $6$. By executing the algorithm over Magma, we enumerate those curves for $p$ up to $1000$. We also succeeded in finding a s.sp.\ hyperelliptic curve of genus $4$ in every $p$ with $p \equiv 2 \pmod{3}$. As a theoretical result, we classify hyperelliptic curves of genus $4$ in terms of automorphism groups in the appendix.