The Supersingularity of Hurwitz Curves

TL;DR

This paper characterizes when Hurwitz curves are supersingular over finite fields, linking supersingularity to specific congruence conditions on the characteristic, and provides a classification for low genus curves.

Contribution

It establishes a precise criterion for supersingularity of Hurwitz curves based on modular conditions and offers a complete classification for small genus and characteristic.

Findings

Supersingularity occurs iff a certain power of p satisfies a congruence condition.

Supersingular Hurwitz curves are also maximal over quadratic extensions.

Complete classification for genus less than 5 and characteristic under 37.

Abstract

We study when Hurwitz curves are supersingular. Specifically, we show that the curve $H_{n,\ell}: X^nY^\ell + Y^nZ^\ell + Z^nX^\ell = 0$, with $n$ and $\ell$ relatively prime, is supersingular over the finite field $\mathbb{F}_{p}$ if and only if there exists an integer $i$ such that $p^i \equiv -1 \bmod (n^2 - n\ell + \ell^2)$. If this holds, we prove that it is also true that the curve is maximal over $\mathbb{F}_{p^{2i}}$. Further, we provide a complete table of supersingular Hurwitz curves of genus less than 5 for characteristic less than 37.