Families of p-adic modular forms

Northwestern University, Evanston IL

Investigators

Abstract

Abstract for the proposal of Calegari DMS-0701048 This project investigates the relationship between Galois representations and automorphic forms using tools from homology, commutative algebra, and p-adic analysis. Of particular interest are modular forms that are of cohomological type but are not associated to Shimura varieties, especially modular forms over an imaginary quadratic field K. 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. A long-term goal of the project is to establish the modularity of elliptic curves over K, adapting the method of Taylor-Wiles to this setting. Modular forms over K 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 has been intensely studied in relation to such questions as Thurston's "Virtual Positive Betti Number Conjecture." However, in number theory, these cohomology classes have conjectural associations to motives and Galois representations. The tension between these different perspectives makes this a fertile area for interdisciplinary research. Wiles' famous work on Fermat's last theorem established a correspondence between two classes of disparate objects: elliptic curves, given by polynomial equations of degree three in two variables, and modular forms.Associated to each elliptic curve is a modular form, which is like the DNA of the corresponding elliptic curve: many properties of the elliptic curve can be determined directly from the modular form. An emerging theme in number theory (and beyond) in the last twenty years is the Langlands program, a vast generalization of this correspondence. In the Langlands program, one considers algebraic varieties, which are finite collections of polynomial equations of arbitrary degree, and automorphic forms, which, like modular forms in the classical case, play the role of algebraic variety DNA. For example, given a set of equations, one would like to know if there exist infinitely many solutions where all the variables are integers. Conjecturally, one can determine this directly from the corresponding automorphic form by computing whether a particular integral vanishes. The Langlands program is still, in many respects, in its infancy --- we neither know exactly how to relate systems of equations, in general, to automorphic forms, nor have we "cracked the code" for understanding how to extract useful information from automorphic forms, or have even determined all the components from which automorphic forms are built. Nevertheless, this broad area of study promises to be a guiding problem in number theory for the foreseeable future.