Automorphic Galois Representations and Automorphic L-functions

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Number theory originates from the study of solutions to equations in whole numbers. It is one of the oldest branches of mathematics, and its methods have for millenia been based on the interaction between the divisibility properties of whole numbers and their size. Twentieth-century number theory formalized these two properties in two different ways. Divisibility and the measure of size can be seen both as geometric and as dynamical properties, the latter rooted in the equations of mathematical physics. The branch of mathematics concerned with their geometric relations is called arithmetic geometry; the branch concerned with their dynamical relations is called automorphic forms. Symmetry plays a central role in both arithmetic geometry and automorphic forms; the hypothetical Langlands correspondence unifies these two branches by showing how each kind of symmetry encodes the other. The PI proposes to use this coding to understand objects on one side of the correspondence in terms of properties of the corresponding object on the other side. A particular focus is the transfer of divisibility properties of automorphic forms to arithmetic geometry, which often leads to surprisingly precise information about solutions of equations.The project is a contribution to the arithmetic theory of automorphic forms, in the setting of the Langlands program, with special attention to the arithmetic of motives and their associated Galois representations arising in the cohomology of Shimura varieties, directly or by application of congruence methods. The long-term goals are the identification of all such motives and all such Galois representations (the modularity problem) and the proof of outstanding conjectures on the arithmetic of motives, notably Deligne's conjecture on special values of L-functions, and the conjectures of Greenberg, Coates, Perrin-Riou, and others on the existence of p-adic L-functions, for the motives obtained in this way. Special attention is given to special values of tensor product L-functions. This project fits into an international program to use the full range of available techniques to extend to all such motives results established for L-functions of elliptic modular forms; the PI is actively collaborating with colleagues in France, Austria, Israel, Japan, and Hong Kong, as well as in the United States and Canada. The methods involved in the present project combine standard techniques from arithmetic geometry and automorphic forms, the differential-geometric approach to cohomological automorphic forms on which the PI has worked for many years, with p-adic analysis, and representation theory.