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. The principal investigator and his collaborators are investigating a possible generalization of Serre's conjecture to the context of Hilbert modular forms and Galois groups over totally real fields. The prediction of the weights suggests interesting new phenomena, and the immediate aims of the research are to investigate these phenomena and provide numerical evidence for the generalization.
Serre's conjecture and its generalizations can be viewed as part of Langlands' program, which 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 investigator and collaborators, building on Wiles' work on Fermat's Last Theorem, proved that all rational elliptic curves arise from modular forms), 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.
