Some Problems in Geometry of Shimura Varieties (SPGSV)
Some Problems in Geometry of Shimura Varieties
Date du début: 1 oct. 2012,
Date de fin: 30 sept. 2017
"The Andre-Oort conjecture is an important problem in the theory of Shimura varieties. It also has significant applications in other areas of Number Theory, such as transcendence theory. The conjecture was proved assuming the Generalised Riemann Hypothesis by Klingler, Ullmo and Yafaev. Very recently, Jonathan Pila came up with a very promising strategy for proving the Andre-Oort conjecture unconditionally. The first main aim of this proposal is to combine Pila's ideas with the ideas of Klingler-Ullmo-Yafaev in order to obtain a proof of the Andre-Oort conjecture without the assumption of the GRH. We then propose to use these methods to attack the Zilber-Pink conjecture, a very vast generalisation of Andre-Oort. We also propose to consider several problems closely related to geometry of Shimura Varieties and the Andr\'e-Oort conjecture. Namely Coleman's cponjecture on finiteness of the number of Jacobians with complex multiplication for curves of large genus, the Mumford-Tate conjecture on Galois representations attached to abelian varieties over number field and Lang's conjecture on rational points on hyperbolic varieties in the context of Shimura varieties."
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