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

TL;DR

This paper develops an algorithm to enumerate superspecial hyperelliptic curves of genus 5 with specific automorphism groups, successfully applying it to characteristics between 12 and 999, advancing understanding in higher genus cases.

Contribution

The paper introduces a feasible algorithm for enumerating certain superspecial genus 5 hyperelliptic curves with large automorphism groups, filling a gap in higher genus curve classification.

Findings

Successfully enumerated superspecial curves in characteristics 12 to 999.

Algorithm implemented in Magma demonstrates practical enumeration.

Advances understanding of superspecial curves of genus 5 with specific automorphism groups.

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 $11 < p < 1000$.