Low Genus Curves on Compact Shimura Surfaces

We investigate and compare the behaviors of complex curves on two types of compact Shimura surfaces: compact variants of Picard and Hilbert modular surfaces. For the compact variant of Picard modular surfaces, we prove there are no curves of a fixed genus when the defining discriminant is sufficiently large. For the compact variant of Hilbert modular surfaces, we prove there are no nonspecial rational and elliptic curves when the defining discriminant is sufficiently large, but rational and elliptic special curves can persist. These can be viewed as special cases of complex function field analogs of uniform boundedness on the endomorphism types of abelian varieties.