The supersingularity of Hurwitz curves

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

doi:10.2140/involve.2019.12.1293

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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

DOI: 10.2140/involve.2019.12.1293

Abstract

We study when Hurwitz curves are supersingular. Specifically, we show that the curve $H_{n, ℓ} : X^{n} Y^{ℓ} + Y^{n} Z^{ℓ} + Z^{n} X^{ℓ} = 0$ , with $n$ and $ℓ$ relatively prime, is supersingular over the finite field $F_{p}$ if and only if there exists an integer $i$ such that $p^{i} \equiv - 1 mod (n^{2} - n ℓ + ℓ^{2})$ . If this holds, we prove that it is also true that the curve is maximal over $F_{p^{2 i}}$ . 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

Milestones

Received: 15 November 2018

Revised: 24 June 2019

Accepted: 6 July 2019

Published: 25 October 2019

Communicated by Ken Ono

Authors

Erin Dawson
Colorado State University Fort Collins, CO United States

Henry Frauenhoff
Colorado State University Fort Collins, CO United States

Michael Lynch
Colorado State University Fort Collins, CO United States

Amethyst Price
Colorado State University Fort Collins, CO United States

Seamus Somerstep
Colorado State University Fort Collins, CO United States

Eric Work
Colorado State University Fort Collins, CO United States

Dean Bisogno
Colorado State University Fort Collins, CO United States

Rachel Pries
Colorado State University Fort Collins, CO United States

Vol. 12, No. 8, 2019