Invariants of Some Algebraic Curves Related to Drinfeld Modular Curves

Abstract

We collect some facts about Drinfeld modular curves for a polynomial ring Fq[T] over a finite field Fq. These include formulas for the genera, the numbers of cusps and elliptic points, descriptions of the function fields and fields of definition, and other rationality properties. We then show that any series of Drinfeld modular curves of Hecke type X0(Nk), where Nk∈Fq[T] is coprime with T and deg(Nk)→∞, gives rise to an asymptotically optimal series of curves over Fq2.