Arithmetic of modular forms

Brandeis University, Waltham MA

Investigators

Abstract

The principal investigator and his collaborators propose to work on aspects of Langlands' program and related topics. A conjecture of Serre predicts that certain 2-dimensional mod p representations of Galois groups over Q arise from modular forms, and it further predicts the level and weight of the form. Serre's conjecture can be viewed as a mod p version of Langlands' correspondence, and the weight part as a manifestation of its local behavior at p. A main focus of the proposed research concerns a generalization of the weight part of Serre's conjecture to the context of Hilbert modular forms and totally real fields, where the developing formulation reveals new phenomena. The research also addresses topics to which this has connections and applications, including congruences between modular forms, deformations of Galois representations, the global Langlands' correspondence and special values of L-functions. Langlands' program predicts a deep correspondence between objects from algebraic geometry (solution sets of polynomial equations, such as elliptic curves) and objects from representation theory (functions with symmetry properties, such as modular forms). While there has been significant recent progress (for example, the advances of Wiles and of Lafforgue), Langlands' program remains largely conjectural. The links it provides between two seemingly different branches of mathematics reveal properties of the integers, which can, in turn, have applications to cryptography.