The modular forms on the quotient $\mathrm{SL}_2(\mathbb{Z})\backslash \mathcal{H}$ can be viewed as $\mathrm{SL}_2(\mathbb{Z})$-invariant holomorphic differentials on $\mathcal{H}$. Interpreting $\mathrm{SL}_2(\mathbb{Z})\backslash \mathcal{H}$ as the moduli space of elliptic curves, these forms can equivalently be described as global sections of the Hodge line bundle. A natural question is whether this perspective extends beyond $\mathrm{SL}_2(\mathbb{Z})$.

In this talk, I will introduce modular forms on certain Shimura varieties and illustrate the definitions through a sequence of examples: Hilbert modular surfaces, unitary Shimura curves, and finally a unitary Shimura surface arising from a special family of cyclic covers of $\mathbb{P}^1$. I will explain how the geometry of this family makes the Hodge line bundle computable, and how level structure on the Shimura variety can be interpreted concretely in this setting.