The supersingularity of Hurwitz curves
| Autor: | Michael J. Lynch, Dean Bisogno, Seamus Somerstep, Eric Work, Erin Dawson, Henry Frauenhoff, Amethyst Price, Rachel Pries |
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| Rok vydání: | 2019 |
| Předmět: |
14H40
Pure mathematics 11E81 Mathematics - Number Theory minimal curve Fermat curve 11G20 14H45 Mathematics::Number Theory 11G10 General Mathematics 11G20 11M38 14H37 14H45 11E81 (Primary) 11G10 14H40 14K15 (Secondary) 11M38 Mathematics::Algebraic Geometry Hasse–Weil bound supersingular curve FOS: Mathematics Hurwitz curve 14H37 Number Theory (math.NT) 14K15 maximal curve Mathematics |
| Zdroj: | Involve 12, no. 8 (2019), 1293-1306 |
| ISSN: | 1944-4184 1944-4176 1293-1306 |
| DOI: | 10.2140/involve.2019.12.1293 |
| Popis: | 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. 15 pages. Accepted to Involve |
| Databáze: | OpenAIRE |
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