Residual Albanese quotients of Picard modular surfaces, and rational points

by Prof. Dinakar Ramakrishnan (California Institute of Technology, USA)

Thursday 30 Aug 2012, 16:00 → 17:00 Asia/Kolkata

AG-69 (Colaba Campus)

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.