Joint IAS/Princeton University Number Theory Seminar

This is a report on a work in progress in collaboration with Dinakar Ramakrishnan. A celebrated result of Mazur states that open modular curves of large enough level do not have rational points. We study analogous questions for the Picard modular surfaces, which are Shimura varieties for a unitary group in 3 variables defined over an imaginary quadratic field $M$. For each $M$ we provide specific levels for which those surfaces are Mordellic, i.e. have only finitely many points over any number field, and we have a project aiming at showing the non-existence of $M$-rational points at deep enough level.