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

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 $C_p:x^3+y^3+w^3=2yw+z^2=0$ in the projective $3$-space with $p>2$ is a superspecial curve of genus $4$ if and only if $p\equiv 2(\mathrm{mod}3)$. Our computational results show that $C_p$ with $p\equiv 2(\mathrm{mod}3)$ are maximal curves over ${\mathbb{F}}_{p^2}$ for all $3\le p\le 269$.

An article putting results in this paper and in [arXiv:1807.04394] together will be appeared in Communications in Algebra

References in corpus (1)

• Superspecial curves of genus $4$ in small characteristic

Cited by in corpus (1)

• The existence of supersingular curves of genus 4 in arbitrary characteristic