Local to Global Compatibility, p-adic Local Langlands and p-adic Level Lowering/Raising

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"This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5)."Coleman and Paulin first intend to study the consequences of the failure of local to global compatibility (away from p) on the eigencurve. Its failure should be intimately related to the local geometry of the eigencurve. In particular it's failure should correspond to crossings between special and principal series components. This is a a p-adic geometric analogue of Ribet's level raising and level lowering theorems. As a second project they intend to establish a local to global principle at p by developing the theory of p-adic modular forms using Fontaine's notion of (phi, gamma)- modules and Colmez's p-adic local langlands correspondence. This may help to suggest a more conceptual definition for a p-adic automorphic representation, at least for GL2.The classical Langlands philosophy suggests that there is a profound duality between real analysis, arithmetic and geometry. The glue binding these different mathematical realms is representation theory. Langlands original insight was that there should be a natural corespondence between arithmetic data (Galois representations) and real analytic data (automorphic representations), mediated in many cases by geometry. The galois representations coming from geometry and therefore conjecturally automorphic have strong restrictions - there are many which cannot be geometric. In recent years it has become apparent that such representations should correspond to some p- adic analogue of an automorphic representation. Coleman and Paulin intend to study this new p-adic correspondence in the case of modular forms. This will help to suggest a more conceptual approach to the rapidly emerging p-adic langlands philosophy.