Deligne-Lusztig curves as ray class fields
type: Item
instance of: scholarly article
title: Deligne-Lusztig curves as ray class fields
zbMATH Open document ID: 0919.11076
published in: Manuscripta Mathematica
publication date: 1999-08-16T00:00:00Z
full work available at URL: http://hdl.handle.net/2027.42/42036
review text: For a finite field \(K\) of square order \(q^2\), the Hermitian curve over \(K\) is the smooth projective curve corresponding to the function field \(K(x, y)\) with defining equation \(y^q+y=x^q\). The Ree and Suzuki curves, which are defined over finite fields \(K\) of non-square order in characteristic 2 and 3, have similar explicit definitions. All these curves are Deligne-Lusztig curves associated to certain linear algebraic groups, and they are `optimal' in the sense that their number of points over \(K\) is maximal with respect to the genus. The author shows that in all three cases, the function field of the curve has an easy description in terms of class field theory: it is the maximal abelian extension of \(K(x)\) of conductor \(k\cdot(\infty)\) in which all places different from the infinite place \((\infty)\) split completely. Here \(k\) is some explicit integer.
Mathematics Subject Classification ID: 11R37
Mathematics Subject Classification ID: 11R58
Mathematics Subject Classification ID: 20G40
zbMATH DE Number: 1251326
zbMATH Keywords: ray class field
zbMATH Keywords: optimal curve
zbMATH Keywords: finite field
zbMATH Keywords: function field
zbMATH Keywords: Ree curves
zbMATH Keywords: Suzuki curves
zbMATH Keywords: Deligne-Lusztig curves
zbMATH Keywords: linear algebraic groups
author: Kristin Lauter
reviewed by: Peter Stevenhagen
MaRDI profile type: Publication
OpenAlex ID: W2024688117
DOI: 10.1007/S002290050127
description: scientific article; zbMATH DE number 1251326