Analytic Continuation of p-adic automorphic forms and applications to the Langlands program

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The overarching aim of this research proposal is to use methods of p-adic analysis and p-adic geometry to make important advances in the p-adic Langlands Program. Much of our work will be devoted to making bridges between p-adic and classical Langlands programs, enabling us to make important progress in the classical Langlands program using p-adic analytic methods. The methodology is largely through analytic and algebraic geometry and a main underlying technique will be that of p-adic analytic continuation of automorphic forms. The technique of p-adic analytic continuation of automorphic forms first emerged in the groundbreaking work of Buzzard-Taylor. After Buzzard-Taylor's work, we invented a method for proving classicality criteria using p-adic analytic continuation which established analytic continuation as an indispensable tool in the theory of p-adic automorphic forms. Our method, though our work and others', has produced many classicality criteria, some of which had been missing from the literature for too long, despite the availability of other aspects of the theory. One of the main long term goals of our proposal is to prove classicality criteria in the many cases that remain open despite recent flurry of progress. In fact, one strand of research in our proposal is devoted to the study of various analytic continuation techniques and direct applications. Under this umbrella, another long term objective is to study domains of automatic analytic continuation for p-adic automorphic forms, a notion that has played a crucial role in our recent proof of the Artin conjecture over certain totally real fields. Under this strand of research, a major objective is the study of directional classicality and integrability introduced by Breuil which have applications to the ongoing effort towards a p-adic local Langlands correspondence beyond the case of GL_2(Q_p). Another strand of research in our proposal is inspired by our proof of the Artin conjecture. In 1999, Buzzard and Taylor proved modularity of Galois representations in weight one using p-adic modular forms. Their work led to a proof of many cases of the classical Artin conjecture. After more than 15 years of resistance, this work has been recently generalized by us to the case of totally real fields. Our work and its subsequent generalizations by Pilloni, Sasaki, Stroh, and Tian have almost settled the Artin conjecture over totally real fields. Our long term objective here is to use our knowledge of p-adic automorphic forms and their analytic continuation to prove automorphy of Galois representation in low weights. This encompasses a vast possibility of research generalizing our work on the Artin conjecture over totally real fields. We intend to focus our attention to the case of partial weight one Hilbert modular forms. We also plan to study a mod-p version of this phenomenon which would provide many remaining cases of the refined conjecture of Serre over totally real fields. An essential feature of our work will be to view p-adic automorphic forms as sections of vector bundles over Shimura varieties: this avails us of powerful techniques in algebraic and analytic geometry. For example, Buzzard-Taylor's approach and our generalization of it are feasible only through this geometric interpretation. A major strand of research in our proposal is, hence, devoted to the study of mod-p and p-adic Geometry of Shimura varieties. We plan to study stratifications on the mod-p Shimura varieties of interest from angles that are guided by our study of p-adic analytic continuation of p-adic automorphic forms. These results will be then used to study the p-adic geometry of Shimura varieties and the p-adic dynamics of the U_p operators on them.