The Supersingularity of Hurwitz Curves
| Autor: | Bisogno, Dean, Dawson, Erin, Frauenhoff, Henry, Lynch, Michael, Price, Amethyst, Pries, Rachel, Somerstep, Seamus, Work, Eric |
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| Rok vydání: | 2018 |
| Předmět: | |
| Zdroj: | Involve 12 (2019) 1293-1306 |
| Druh dokumentu: | Working Paper |
| 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. Comment: 15 pages. Accepted to Involve |
| Databáze: | arXiv |
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