Rational points on Picard modular surfaces

Picard modular surfaces X, which are smooth compactifications of the quotients Y of the complex ball by a discrete subgroups \Gamma of SU(2,1), have been studied from various points of view. They are often defined over an imaginary quadratic field M, and we are interested in the rational points of X over finite extensions k of M. In a joint work with M. Dimitrov we deduced some finiteness results for the points on the open surfaces Y of congruence type with a bit of level (using theorems of Faltings, Rogawski and Nadel), which we will first recall.A recent result we have is an analogue, for polarized abelian threefolds with multiplication by O_M and without CM factors, an analogue of the classical theorem of Manin asserting that for p prime, there is a universal r=r(p,k) such that for any non-CM elliptic curve E, the p^r-division subgroup of E has no k-rational line. If time permits, we will explain the final objective of our ongoing program.