Arithmetic of Shimura varieties and automorphic forms and Applications – ArShiFo

In recent years, p-adic methods (Fontaine Theory, Galois deformations) and geometric methods (deformations of varieties, local methods for Shimura varieties) have been used successfully to solve central problems of modern Number Theory. Let us mention

-Serre's modularity conjecture, proven by Khare and Wintenberger,

-the p-adic Langlands correspondence for GL_2(Q_p), proven by Colmez,

-Fontaine-Mazur's conjecture for GL_2 over Q, proven by Kisin,

-Sato-Tate conjecture, proven by Clozel-Harris-Shepherd-Barron-Taylor and Taylor,

-the Fundamental Lemma for unitary groups, by Ngô, Laumon-Ngô and Chaudouard-Laumon for the weighted version

-the vanishing of the mu-invariant of certain p-adic L functions, by Hida,

It is reasonable to believe that these methods will yield new results in a near future. In fact the importance of the subject can be verified by noting that nine out of ten of the invited speakers in Number Theory at the 2010 ICM work on these questions, that Clay Institute organized a four week-long summer school on the p-adic aspects of this subject in 2009 whereas the IAS organizes a special year on these methods in 2010-2011 and the Fields Institute in 2011-2012. The goal of the present project is to provide protagonists of the field, french or working in France, with the best conditions to pursue their effort of
research. Let us mention that Ngô Bao-Chau has accepted a special chair of Visiting Professor at Paris 13, of one month per year for the duration of the project (2010-2014) .

In the framework of this project, we plan to pursue and develop the Paris13-Columbia video-seminar (adjoining also UCLA) , and to organize two conferences, one in 2012 at the University of Paris 13 et l'autre, de plus grande ampleur, en 2014 à Luminy).
The financial support demanded is for financing of these conferences, for funding a three-year doctoral position and a one-year posdoctoral position at Paris 13, a one-year post-doctoral position at Ecole Polytechnique; the applicants will be selected among the best available in France and abroad. We also plan to cover missions; for instance, we hope to help project members (including our doctoral students) to participate to activities of the special year a t IAS (R. Taylor), at Fields Institute (Emerton-Darmon), at the LMS conference in Durham in 2011 (Diamond), at the Banff conference in 2011 (Moeglin, Vatsal) on the consequences of Arthur's book, the UCLA conference in 2012 for Hida's 60th birthday (Khare), and others.