A note on certain superspecial and maximal curves of genus $5$

TL;DR

This paper characterizes specific maximal superspecial genus 5 curves over finite fields, identifying conditions for their maximality and superspeciality, and introduces new families of such curves with broader types.

Contribution

It provides a precise characterization of certain maximal superspecial genus 5 curves and introduces new families of maximal curves of more general types.

Findings

Desingularization $T_p$ is maximal superspecial if and only if $p ot\equiv 0, 1, 4, 11, 14$ mod 15.

Conditions for maximality depend on congruences of the characteristic prime.

New families of maximal curves of more general type are constructed.

Abstract

In this note, we characterize certain maximal superspecial curves of genus $5$ over finite fields. Specifically, we prove that the desingularization $T_p$ of $x y z^3 + x^5 + y^5 = 0$ is a maximal superspecial trigonal curve of genus $5$ if and only if $p \equiv -1 \pmod{15}$ or $p \equiv 11 \pmod{15}$. Moreover, we give families of maximal curves of more general type, which include $T_p$.