The Geometry of Drinfeld Modular Forms

Abstract

We give a geometric perspective on the algebra of Drinfeld modular forms for congruence subgroups $\Gamma\leq \GL_2(\bbF_q[T]).$ In particular, we describe an isomorphism between the section ring of a line bundle on the stacky modular curve for $\Gamma_2$ and the algebra of Drinfeld modular forms for $\Gamma_2,$ where $\Gamma_2$ is the subgroup of square-determinant matrices in $\Gamma.$ This allows one to compute the latter ring by geometric invariants using the techniques of Voight, Zureick-Brown and O'Dorney. We also show how to decompose the algebra of modular forms for $\Gamma_2$ into a direct sum of two algebras of modular forms for $\Gamma$ and generalize this result to a larger class of congruence subgroups.