Arithmetic and geometry of Picard modular surfaces

Of interest are (i) the conjecture of Bombieri (and Lang) that for any smooth projective surface $X$ of general type over a number field $k$, the set $X(k)$, of $k$-rational points is not Zariski dense, and (ii) the conjecture of Lang that $X(k)$, is even finite if in addition $X$ is hyperbolic, i.e., there is no non-constant holomorphic map from the complex line $C$ into $X(C)$. We can verify them for the Picard modular surfaces $X$ which are smooth toroidal compactifications of congruence quotients $Y$ of the unit ball in $\mathbb C^2$. We will describe an ongoing program, with Mladen Dimitrov, to prove moreover that for suitable deep levels, $Y$ has no rational points over the natural field of definition $k$. We use the theory of automorphic forms on the associated unitary group in three variables, and some geometry, to adapt and develop an analogue of the elegant method of Mazur proving such a result for modular curves (in 1977). We use the residual and cuspidal quotients of the Albanese variety of $X$, and a suitable product of such provides a replacement for Mazur's Eisenstein quotient. If time permits, we will also explain the connection to the problem of uniform boundedness of torsion for principally polarized abelian threefolds $A$ with multiplication by the ring of integers of an imaginary quadratic field.