**Dinakar Ramakrishnan**

California Institute of Technology, USA

August 30, 2012

**Residual Albanese quotients of Picard modular surfaces, and rational points**:
A well known result of Mazur shows the paucity of rational points outside the cusps on modular curves of high genus. This talk will present a program with M. Dimitrov to try to establish a weak analogue for the Picard modular surfaces $X$, which arise as quotients of the unit ball in $C^2$ by congruence subgroups of $U(2,1)$ associated to an imaginary quadratic field $E$. It is
known that the Albanese variety of any such $X$ is of $CM$ type. A key role for us will be played by the part of $Alb(X)$ coming from residual automorphic forms on $U(2,1)$. The presentation will be concrete, presenting examples of residual quotients $A$ of finite Mordell-Weil group, and will investigate consequences for the arithmetic on $X$.