#### Vol. 12, No. 8, 2019

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The supersingularity of Hurwitz curves

### Erin Dawson, Henry Frauenhoff, Michael Lynch, Amethyst Price, Seamus Somerstep, Eric Work, Dean Bisogno and Rachel Pries

Vol. 12 (2019), No. 8, 1293–1306
##### Abstract

We study when Hurwitz curves are supersingular. Specifically, we show that the curve ${H}_{n,\ell }:{X}^{n}{Y}^{\ell }+{Y}^{n}{Z}^{\ell }+{Z}^{n}{X}^{\ell }=0$, 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 ${p}^{i}\equiv -1\phantom{\rule{0.2em}{0ex}}mod\phantom{\rule{0.2em}{0ex}}\left({n}^{2}-n\ell +{\ell }^{2}\right)$. 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.

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##### Keywords
Hurwitz curve, Hasse–Weil bound, maximal curve, minimal curve, Fermat curve, supersingular curve
##### Mathematical Subject Classification 2010
Primary: 11G20, 11M38, 14H37, 14H45, 11E81
Secondary: 11G10, 14H40, 14K15