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 $C_p : x^3+y^3+w^3= 2 y w + z^2 = 0$ 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 $3 \leq p \leq 269$.