Algorithmic study of superspecial hyperelliptic curves over finite fields

TL;DR

This paper develops algorithms to enumerate superspecial hyperelliptic curves over finite fields and to compute their automorphism groups, enabling classification and analysis of these curves for specific genera and field sizes.

Contribution

It introduces new algorithms for enumerating superspecial hyperelliptic curves and computing automorphism groups, with practical implementation and applications to genus 4 curves.

Findings

Enumerated superspecial hyperelliptic curves of genus 4 over various finite fields.

Identified maximal and minimal hyperelliptic curves among the enumerated ones.

Provided algorithms with proven applicability under certain conditions.

Abstract

This paper presents algorithmic approaches to study superspecial hyperelliptic curves. The algorithms proposed in this paper are: an algorithm to enumerate superspecial hyperelliptic curves of genus $g$ over finite fields $\mathbb{F}_q$, and an algorithm to compute the automorphism group of a (not necessarily superspecial) hyperelliptic curve over finite fields. The first algorithm works for any $(g,q)$ such that $q$ and $2g+2$ are coprime and $q>2g+1$. As an application, we enumerate superspecial hyperelliptic curves of genus $g=4$ over $\mathbb{F}_{p}$ for $11 \leq p \leq 23$ and over $\mathbb{F}_{p^2}$ for $11 \leq p \leq 19$ with our implementation on a computer algebra system Magma. Moreover, we found maximal hyperelliptic curves and minimal hyperelliptic curves over $\mathbb{F}_{p^2}$ from among enumerated superspecial ones. The second algorithm computes an automorphism as a concrete element in (a quotient of) a linear group in the general linear group of degree $2$.