Cohomology of Arithmetic Groups and Galois Representations

Boston College, Chestnut Hill MA

Investigators

Abstract

The Principal Investigator (PI), with various collaborators, studies an area of number theory concerned with representations of Galois groups over the rationals and cohomology of finite-index subgroups of GL(n,Z). The link between these is given by the action of Hecke operators on the cohomology and the image of Frobenius elements in the Galois representation. The PI explores the "ADPS" conjecture, where the cohomology and Galois representation are both mod-p valued. He outlines an approach for proving this conjecture in certain 3-dimensional cases, testing it in certain 4-dimensional cases, and independently verifying some of its consequences for Diophantine problems. The PI states a new conjecture that links the mod-p objects with automorphic representations and suggests an approach for proving this conjecture in the 3-dimensional case. Finally, the PI and a colleague continue their study of p-adic families of automorphic cohomology. On the one hand, for certain classes, e.g those on GL(3) not lifted from GL(2), they consider questions of p-adic rigidity. On the other hand, for deformable classes, they investigate 2-variable p-adic L-functions. The ability to solve systems of algebraic equations has been central to modern science and engineering. It has also always been one of the main topics in mathematics, driven both by these applications and by its intrinsic aesthetic appeal. When the equations have whole-number coefficients, their study becomes part of number theory. The theory of prime numbers (numbers without smaller divisors except 1, such as 2,3,5 and 7) is central here. In the last 50 years, applications of these theories have become crucial to cryptography and communications theory. The Principal Investigator studies delicate questions concerning the fine structure of sets of solutions to systems of polynomial equations, the symmetries they exhibit, and their (often surprising) relationship to various interesting geometric and topological objects. These relationships are mediated by how the equations behave with respect to the various prime numbers. There are a number of conjectural explanations of these relationships, and the PI studies them, proving them in certain "easy" cases and verifying them by computer in more complicated cases. He also investigates phenomena which compare whole families of these structures with respect to a fixed prime number.