Overconvergent cohomology of higher rank groups

Trustees Of Boston University, Boston

Investigators

Abstract

In the last five years, a tremendous amount of progress has been made in the p-adic theory of modular forms. Skinner and Urban's proof of the main conjecture, Khare and Winterberger's proof of Serre's conjecture, Emerton's and Kisin's independent work on the Fontaine-Mazur conjecture represent only a partial list of the incredible recent progress that has occurred. Moreover, there is a rich emerging p-adic theory of automorphic forms generalizing the corresponding p-adic theory of modular forms. However, the vast computational methods available in the study of classical modular forms, do not yet exist in the higher rank case. Currently, there is a gap in the understanding of the shape and structures of many of the fundamental objects that appear in this more general setting. Pollack proposes to make a concrete and explicit study of overconvergent cohomology of higher rank groups along the lines that he and Glenn Stevens did in the case of classical modular forms. This previous study had both theoretical and computational aspects that, in particular, led to a deeper understanding of two variable p-adic L-functions and to a p-adic method of computing (conjecturally) global points on elliptic curves. He hopes that a similar study in the higher rank case will lead to a better understanding of analogous phenomena in this more general setting. For over a hundred years, mathematicians have studied modular forms which are certain complex analytic functions possessing many symmetries. The definition of these objects is through analysis (the rigorous study of calculus); nonetheless, these objects have made their way into the center of number theory as they possess a wealth of arithmetic information of the type that would have interested Euclid, Gauss, Fermat, etc. A deep understanding of these arithmetic properties of modular forms was one of the key steps in Andrew Wiles' proof of Fermat's Last Theorem. Moreover, the theory of automorphic forms -- a far-reaching generalization of modular forms -- has proven to also be extremely rich arithmetically. However, unlike classical modular forms where one can compute with relative ease, general automorphic forms can be extremely difficult to compute and experiment with numerically. Pollack proposes to make an in depth study of the computation of certain kinds of automorphic forms with the hope of gaining a better arithmetic understanding of them through conjecture and numerical exploration.