Producing supersingular curves of genus five

TL;DR

This paper proves the existence of supersingular genus five curves in characteristic p for primes p ≡ 3 mod 4, constructs such curves as double covers, and conjectures a similar result for p ≡ 1 mod 4, supported by computational evidence.

Contribution

It establishes the existence of supersingular genus five curves in certain characteristics and introduces a conjecture extending this to other primes, supported by computational verification.

Findings

Existence of supersingular genus five curves for p ≡ 3 mod 4.

Construction of these curves as unramified double covers.

Computational evidence supporting the conjecture for p < 100.

Abstract

For a prime $p$ congruent to three modulo four, we prove that there exists a smooth curve of genus five in characteristic $p$ that is supersingular. We produce this curve as an unramified double cover of a curve of genus three. We conjecture that the setting of unramified double covers of curves of genus three also produces supersingular curves of genus five when $p$ is congruent to one modulo four, and we computationally verify this conjecture for primes less than $100$. These results can be viewed as a generalization of work of Ekedahl and of Harashita, Kudo, and Senda.