On the existence of superspecial nonhyperelliptic curves of genus $4$

TL;DR

This paper proves the existence of superspecial nonhyperelliptic curves of genus 4 over infinitely many primes, providing an explicit family of such curves and demonstrating their maximality over certain finite fields.

Contribution

It offers an elementary proof of the existence of these curves for infinitely many primes and characterizes them explicitly via a specific algebraic variety.

Findings

Existence of superspecial nonhyperelliptic genus 4 curves for infinitely many primes p.

The variety C_p is superspecial if and only if p ≡ 2 mod 3.

C_p are maximal curves over p^2 for 3  p  269.

Abstract

A curve over a perfect field $K$ of characteristic $p > 0$ is said to be superspecial if its Jacobian is isomorphic to a product of supersingular elliptic curves over the algebraic closure $\overline{K}$. In recent years, isomorphism classes of superspecial nonhyperelliptic curves of genus $4$ over finite fields in small characteristic have been enumerated. In particular, the non-existence of superspecial curves of genus $4$ in characteristic $p = 7$ was proved. In this note, we give an elementary proof of the existence of superspecial nonhyperelliptic curves of genus $4$ for infinitely many primes $p$. Specifically, we prove that the variety $C_p : x^3+y^3+w^3= 2 y w + z^2 = 0$ in the projective $3$-space with $p > 2$ is a superspecial curve of genus $4$ if and only if $p \equiv 2 \pmod{3}$. Our computational results show that $C_p$ with $p \equiv 2 \pmod 3$ are maximal curves over $\mathbb{F}_{p^2}$ for all $3 \leq p \leq 269$.