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arXiv · 1105.3952

The Automorphism Groups of a Family of Maximal Curves

Robert Guralnick·Beth Malmskog·Rachel Pries
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Abstract

The Hasse Weil bound restricts the number of points of a curve which are defined over a finite field; if the number of points meets this bound, the curve is called maximal. Giulietti and Korchmaros introduced a curve C_3 which is maximal over F_{q^6} and determined its automorphism group. Garcia, Guneri, and Stichtenoth generalized this construction to a family of curves C_n, indexed by an odd integer n greater than or equal to 3, such that C_n is maximal over F_{q^{2n}}. In this paper, we determine the automorphism group Aut(C_n) when n > 3; in contrast with the case n=3, it fixes the point at infinity on C_n. The proof requires a new structural result about automorphism groups of curves in characteristic p such that each Sylow p-subgroup has exactly one fixed point. MSC:11G20, 14H37.

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Robert Guralnick, Beth Malmskog, Rachel Pries. 2011-05-19. The Automorphism Groups of a Family of Maximal Curves. https://arxiv.org/abs/1105.3952

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