The Geometry of Drinfeld Modular Forms

We give a geometric perspective on the algebra of Drinfeld modular forms for congruence subgroups $\mathrm{\Gamma }\le {\text{GL}}_2({\text{bbF}}_q[T])\mathrm{.}$ In particular, we describe an isomorphism between the section ring of a line bundle on the stacky modular curve for ${\mathrm{\Gamma }}_2$ and the algebra of Drinfeld modular forms for ${\mathrm{\Gamma }}_2,$ where ${\mathrm{\Gamma }}_2$ is the subgroup of square-determinant matrices in $\mathrm{\Gamma }\mathrm{.}$ 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 ${\mathrm{\Gamma }}_2$ into a direct sum of two algebras of modular forms for $\mathrm{\Gamma }$ and generalize this result to a larger class of congruence subgroups.