FRG: Collaborative Research: Automorphic forms, Galois representations, periods and p-adic L-functions

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"This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5)."The PIs will study some fundamental problems in algebraic number theory, particularly problems related to the deep links between Galois representations and special values of L-functions (as conjectured in the Birch-Swinnerton-Dyer Conjecture and the Bloch-Kato Conjectures). More specifically, the PIs will study the following themes: (i) Mod-p Galois representations and mod-p modular forms; (ii) Constructing Galois representations and motives associated to automorphic forms; (iii) The Iwasawa Main Conjecture; (iv) p-adic families of automorphic forms and applications; (iv) Algebraic cycles, p-adic L-functions and Euler systems. Thus the main focus will be on p-adic methods in the theory of automorphic forms and Galois representations. The PIs will arrange short-term visits for collaborative research purposes between themselves, affiliated researchers and new comers in the area, be actively involved in graduate training and postdoctoral advising, and organize two workshops and a final conference,with preparation of a proceedings for dissemination of the results.In nontechnical terms, the problems that the PIs will study involve showing the surprising equality of two number-theoretic objects, one defined analytically and the other algebraically. In this way, the problems to be studied are linked by two common philosophical threads: the notion of a reciprocity law, which has a long and deep tradition in number theory, going back to the quadratic reciprocity law of Gauss, and the notion of a class number formula, which goes back to the fundamental ideas of Dirichlet. Further, such equalities of mathematical objects defined in a priori different ways are not just of theoretical interest but tend to have extremely concrete applications, the most striking recent ones being the resolution of Fermat's last theorem and the Sato-Tate conjecture. The workshops, the final conference, and the graduate and post-doctoral advising will have an important impact on the formation of new researchers in the field and on the promotion of new collaborations.