Galois Representations, Monodromy Groups, and Motives

University Of Utah, Salt Lake City UT

Investigators

Abstract

Throughout the mathematical sciences, symmetry is a powerful organizing principle. In number theory, which studies the integers, especially the prime numbers, there are no manifest symmetries of the kind encountered in more intuitive, geometric settings (symmetry of a square under 90-degree rotations, etc.). Nevertheless, much of our deepest knowledge of the primes comes from their connections with the "symmetries" of polynomial equations. This project focuses on the study of Galois representations, which are natural packages for this information about the prime numbers. One of the central programs of modern number theory, initiated by Robert Langlands, proposes new ways to "unpack" Galois representations by relating them to remarkably different mathematical objects arising in geometry. Although they are essentially about the prime numbers, these relations have even had reverberations in fundamental physics. This project involves both constructing new Galois representations and unpacking the information they contain in the light of Langlands' conjectures. This project comprises two research programs. The first will study deformations of Galois representations valued in arbitrary reductive groups, with a view toward establishing in as great generality as possible the existence of geometric p-adic deformations of a given mod p representation. The work is likely to furnish steps toward generalizations of Serre's modularity conjecture to general reductive groups. Such geometric p-adic representations are expected to be the (semi-simple) Galois representations occurring in the cohomology of algebraic varieties over number fields. The second program aims to study a number of questions of Fontaine-Mazur type, seeking to establish the motivic origin of certain Galois representations or, in a more geometric setting, local systems over a curve. The work will require elaborating the motivic structures underlying basic objects of the geometric Langlands program, a problem that should be of long-term independent interest even for the arithmetic Langlands program.