Low Genus Curves on Compact Shimura Surfaces

Abstract

We investigate and compare the behaviors of complex curves on two major types of compact Shimura surfaces: compact variants of Picard and Hilbert modular surfaces. They are quotients of complex hyperbolic balls corresponding to moduli problems for abelian varieties with certain endomorphism types. We prove that if we fix a genus bound, then when the defining discriminant is large enough, low genus curves on the compact variant of Picard modular surfaces do not exist, while low genus curves may persist on the compact variant of Hilbert modular surfaces only if they are special curves, which parametrize those with extra endomorphisms. These results are special cases of a complex function field analogue of uniform boundedness on endomorphism types of abelian varieties. The proof requires understanding the repulsion of special subvarieties and relies on volume estimates in hyperbolic geometry.

arXiv comment48 pages. Major updates compared to the previous version: the case of compact variant of Hilbert modular surfaces is included and it shows different geometric patterns; many of the proofs are upgraded to a more general setting. I will upload a more polished version soon