The investigator and his colleagues study the representation theory of p-adic groups in topological vector spaces over p-adic fields, with the goal of relating such representations to arithmetic by means of a "p-adic analytic local Langlands correspondence." Though still in its early stages, work of the investigator, his collaborator Peter Schneider, and others including Robert Coleman and Barry Mazur, and more recently Matthew Emerton and Christophe Breuil, have provided evidence that such a correspondence exists. The investigator has the hope that this approach will provide a conceptual link between p-adic L-functions, p-adic Galois representations, and p-adic automorphic forms, in the same way that the classical Langlands correspondence does for complex L-functions, automorphic forms, and ell-adic representations.
One principal of current research in number theory is the Langlands program, which proposes a deep relationship between certain complex analytic functions related to algebraic groups called "automorphic forms" and the number of integer solutions to classes of polynomial equations, through the medium of functions called "zeta functions." The power of this idea was strikingly demonstrated by Wiles' proof of Fermat's theorem, which proceeded by establishing one very special case of the Langlands conjectures. A second important principal in this field is the idea that a full understanding of number theory requires the study, not only of the geometric behavior of equations over the real and complex numbers, but also over the less widely known fields of p-adic numbers. Indeed, many of the important classical objects of number theory, such as modular forms andL-functions, that are important in the Langlands project, have p-adic versions. These p-adic versions capture important information not accessible from the classical situation. This project proposes to extend our understanding of the p-adic versions of automorphic forms and zeta functions by extending the philosophy of the Langlands program to the setting of p-adic analysis.
