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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.
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