Compact arithmetic quotients of the complex 2-ball and a conjecture of Lang
Let X be a compact quotient of the unit ball in ℂ^2 by an arithmetic subgroup Γ of a unitary group defined by an anisotropic hermitian form on a three dimensional vector space over a CM field with signature (2,1) at one archimedean place and (3,0) at the others. We prove that if all the torsion elements of Γ are scalar, then X is Mordellic, meaning that for any number field k containing the field of definition of X, the set X(k) of k-rational points of X is finite. The proof applies and combines certain key results of Faltings with the work of Rogawski and the hyperbolicity of X.
Additional InformationWe would like to thank Don Blasius, Laurent Clozel, Najmuddin Fakhruddin, Dick Gross, Haruzo Hida, Barry Mazur, David Rohrlich, Matthew Stover and Shing-Tung Yau for helpful conversations. In fact it was Fakhruddin who suggested our use of Lang's conjecture for abelian varieties. Needless to say, this Note owes much to the deep results of Faltings. 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 the following sources: the Agence Nationale de la Recherche grants ANR-10-BLAN-0114 and ANR-11-LABX-0007-01 for the first author (M.D.), and from the NSF grant DMS-1001916 for the second author (D.R.).
Submitted - 1401.1628v2.pdf