Algorithmic study of superspecial hyperelliptic curves over finite fields

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 $11\le p\le 23$ and over ${\mathbb{F}}_{p^2}$ for $11\le p\le 19$ 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 $2$.