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 Hn, : XnY + Y nZ + ZnX = 0, with n and relatively prime, is supersingular over the finite field Fp if and only if there exists an integer i such that pi 1 mod (n2 n + 2). If this holds, we prove that it is also true that the curve is maximal over Fp2i. Further, we provide a complete table of supersingular Hurwitz curves of genus less than 5 for characteristic less than 37.

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