Arithmetic Quotients of the Complex Ball and a Conjecture of Lang
We prove that various arithmetic quotients of the unit ball in C^n 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.
© 2015 Documenta Mathematica. Received: October 10, 2014. Revised: September 28, 2015. We would like to thank Don Blasius, Jean-François Dat, Najmuddin Fakhruddin, Dick Gross, Barry Mazur, Matthew Stover and Shing-Tung Yau for helpful conversations. In fact it was Fakhruddin who suggested our use of the Mordell-Lang conjecture for abelian varieties. Needless to say, this Note owes much to the deep results of Faltings. In addition, we thank the referee and Blasius for their corrections and suggestions which led to an improvement of the presentation. Thanks are also due to Serge Lang (posthumously), and to John Tate, for getting one of us interested in the conjectural Mordellic property of hyperbolic varieties. Finally, we are also happy to acknowledge partial support from the following sources: the Agence Nationale de la Recherche grants ANR-10-BLAN-0114 and ANR-11-LABX-0007-01 for the first author, and the NSF grant DMS-1001916 for the second author.
Submitted - 1401.1628v4.pdf