Deligne-Lusztig curves as ray class fields
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.
scientific article; zbMATH DE number 1251326