Arithmetic Quotients of the Complex Ball and a Conjecture of Lang

We prove that various arithmetic quotients of the unit ball in Cⁿ are Mordellic, in the sense that they have only finitely many rational points over any finitely generated field extension of Q. In the previously known case of compact hyperbolic complex surfaces, we give a new proof using their Albanese in conjunction with some key results of Faltings, but without appealing to the Shafarevich conjecture. In higher dimension, our methods allow us to solve an alternative of Ullmo and Yafaev. Our strongest result uses in addition Rogawski's theory and establishes the Mordellicity of the Baily-Borel compactifications of Picard modular surfaces of some precise levels related to the discriminant of the imaginary quadratic fields.

2010 Mathematics Subject Classification: 11G18, 14G05, 22E40.

Keywords and Phrases: Rational points; Picard modular surfaces; Albanese varieties; Automorphic representations of unitary groups.

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