The automorphism groups of a family of maximal curves

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 Korchmáros introduced a curve which is maximal over and determined its automorphism group. Garcia, Guneri, and Stichtenoth generalized this construction to a family of curves , indexed by an odd integer , such that is maximal over . In this paper, we determine the automorphism group when ; in contrast with the case , it fixes the point at infinity on . The proof requires a new structural result about automorphism groups of curves in characteristic such that each Sylow -subgroup has exactly one fixed point.

哈塞 - 韦伊界限制了在有限域上定义的曲线的点数；如果点数达到这个界，该曲线被称为极大的。朱列蒂和科尔奇马罗斯引入了一条在\({\mathbb{F}}_{q^{6}}\)上是极大的曲线，并确定了它的自同构群。加西亚、居内里和施蒂希滕托斯将这种构造推广到一族曲线\({\mathcal{C}}_{n}\)，由奇数整数\(n\)索引，使得\({\mathcal{C}}_{n}\)在\({\mathbb{F}}_{q^{2n}}\)上是极大的。在本文中，我们确定当\(n = 3\)时的自同构群\({\rm Aut}({\mathcal{C}}_{3})\)；与\(n = 1\)的情况相反，它固定\({\mathcal{C}}_{3}\)上的无穷远点。该证明需要一个关于特征\(p\)下曲线的自同构群的新的结构结果，即每个西罗\(p\)-子群恰好有一个不动点。