Invariants of some algebraic curves related to Drinfeld modular curves

Abstract

We collect some facts abaut Drinfeld modular curves for a polynomial ring \mathbb{F}_{q}[T] over a finite field \mathbb{F}_{q}. 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 X_{0}(N_{k}), where N_{k}\in\mathbb{F}_{q}[T] is coprime with T and \mbox{deg}(N_{k})\longrightarrow\infty, gives rise to an asymptotically optimal series of curves over \mathbb{F}_{q^{2}}.