Hyperbolicity, automorphic forms and Siegel modular varieties

TL;DR

This paper investigates the hyperbolic properties of compactified quotients of symmetric domains by arithmetic groups, with applications to Siegel modular varieties and the non-existence of certain abelian variety structures.

Contribution

It establishes hyperbolicity results for these quotients and improves previous findings on level structures of abelian varieties over complex function fields.

Findings

Compactifications are Kobayashi hyperbolic modulo boundary

Enhanced previous results on non-existence of certain level structures

Applied techniques to Siegel modular varieties

Abstract

We study the hyperbolicity of compactifications of quotients of bounded symmetric domains by arithmetic groups. We prove that, up to an \'etale cover, they are Kobayashi hyperbolic modulo the boundary. Applying our techniques to Siegel modular varieties, we improve some former results of Nadel on the non-existence of certain level structures on abelian varieties over complex function fields.