Arithmetic Quotients of the Complex Ball and a Conjecture of Lang

Mladen Dimitrov, Dinakar Ramakrishnan

Published 2014-01-08, updated 2014-10-11Version 3

We prove that various arithmetic quotients of the unit ball in ${\mathbb C}^n$ are Mordellic, in the sense that they have only finitely many rational points over any given number field. In the previously known case of compact hyperbolic complex surfaces, we give a new proof using their Albanese as well as some key results of Faltings, but without appealing to the Shafarevich conjecture. Our second and strongest result concerns the open Picard modular surfaces $Y$ whose level divides the discriminant of the imaginary quadratic field over which $Y$ is defined. We use the full force of Rogawski's theory in a key computation of the dimension of their Albanese. Our final result solves an alternative of Ullmo and Yafaev, thus showing that, for arbitrary $n$, the arithmetic quotients under consideration are Mordellic, after possibly taking a finite cover.