CAREER: Arithmetic of Cohomological Automorphic Forms

Northwestern University, Evanston IL

Investigators

Abstract

The investigator studies the relationship between Galois representations and automorphic forms using tools from homology, commutative algebra, and group theory, in the context of the Langlands program. A major innovation in the investigator's recent research is the use of non-commutative Iwasawa theory to study p-adically completed cohomology associated to automorphic forms of cohomological type, in particular, those forms that are not necessarily associated to Shimura varieties. One such class of automorphic forms are modular forms over an imaginary quadratic field. In this case, the associated symmetric space quotients are hyperbolic manifolds of real dimension 3, and thus, the study of such forms is not amenable to the usual techniques of algebraic geometry. Modular forms over imaginary quadratic fields can be thought of geometrically as cohomology classes of certain local systems on arithmetic 3-manifolds. From a topological viewpoint, the cohomology of arithmetic 3-manifolds continues to be a subject of intense study. The tension between the number theoretical and topological perspectives makes this a fertile area for interdisciplinary research. The ultimate goal of the investigator's research is to formulate and prove a general reciprocity statement relating all Galois representations to the cohomology of arithmetic groups, generalizing the reciprocity conjecture of Langlands. A more specific goal is to establish conditional modularity theorems over imaginary quadratic fields, adapting the method of Taylor--Wiles. The Langlands reciprocity conjecture predicts a one-to-one correspondence between two classes of disparate objects: Galois representations, which describe certain symmetries of algebraic numbers, and automorphic forms. Associated to each Galois representation is an automorphic form, which is like the DNA of the corresponding Galois representation: many properties of the Galois representation can be determined directly from the automorphic form. One strategy for establishing this correspondence is a counting argument: show that the number of automorphic forms (of any fixed type) equals the number of corresponding Galois representations. Wiles used this strategy to prove a special case of the Langlands reciprocity conjecture, from which Fermat's Last Theorem follows. Wiles' proof relies on certain auxiliary geometric constructions (Shimura varieties) that are not always available, and hence, any general argument requires a more robust method for counting automorphic forms. The investigator proposes such a method: by decomposing automorphic forms into their (mod-p) constituents and then gluing the pieces back together. This process (p-adic completion) relates the old counting problem to questions in topology, in particular, to well-known conjectures of Thurston. The investigator's research along these lines is helping to foster new collaborations between number theorists and low-dimensional topologists, which may lead to advances in several open questions in both fields. The investigator plans to organize several workshops for graduate students and postdocs designed to train young number theorists in this rapidly changing interdisciplinary field.