Curves on Compact Arithmetic Quotients of Hyperbolic 2-ball

TL;DR

This paper investigates the geometry of compact arithmetic quotients of the hyperbolic 2-ball, showing that for large discriminants, these quotients contain no complex curves of fixed genus, using volume estimates and geometric analysis.

Contribution

It establishes the non-existence of certain complex curves on these quotients for large discriminants, advancing understanding of their geometric structure.

Findings

No complex curves of fixed genus exist for large discriminants

Volume estimates are used to analyze the distribution of special subvarieties

Geometry near quotient and cusp singularities is crucial in the proof

Abstract

We study the geometry of the simplest type of compact arithmetic quotients of the hyperbolic 2-ball $\mathbb{B}^2$, which has a moduli interpretation for certain types of abelian varieties of dimension 6 with $\mathcal{O}_F$-endomorphism, where $F$ is a CM extension of a real quadratic field $\mathbb{Q}(\sqrt{D})$. Under mild assumption, we prove that for any fixed $g$, when the defining discriminant $D$ is large, there will be no complex curves of genus $g$ on this type of arithmetic quotients. The proof uses the technique of volume estimates, which requires us to understand the distribution of special subvarieties and the geometry near quotient and cusp singularities.