On Drinfeld modular curves for SL(2)

TL;DR

This paper investigates Drinfeld modular curves associated with Hecke congruence subgroups of SL(2) over function fields, deriving genus formulas and explicit equations for genus one cases.

Contribution

It provides a genus formula for these curves and explicitly computes Weierstrass equations when the genus is one, advancing understanding of their structure.

Findings

Genus formula for Drinfeld modular curves from Hecke subgroups

Explicit Weierstrass equations for genus one curves

Application of combinatorial methods to modular curve analysis

Abstract

We study the Drinfeld modular curves arising from the Hecke congruence subgroups of $\mathrm{SL}_2(\mathbb{F}_q[T])$. Using a combinatorial method of Gekeler and Nonnengardt, we obtain a genus formula for these curves. In cases when the genus is one, we compute the Weierstrass equation of the corresponding curve.