Chomology of Arithmetic Groups and Galois Representations

Boston College, Chestnut Hill MA

Investigators

Abstract

The principal investigator, together with various collaborators, studies the cohomology of arithmetic groups as modules over an algebra of Hecke operators. He considers conjectures connecting (1) mod-p cohomology classes of congruence groups which are eigenclasses for all the Hecke operators with (2) mod-p representations of the Galois (symmetry) group of the algebraic numbers. These conjectures amount to saying there should exist reciprocity laws (usually non-abelian) connecting the two types of objects. The conjectures are tested computationally, which leads to further refinements of them. The principal investigator outlines an approach for proving them in some special cases. Another set of computer calculations probe for the existence of automorphic representations of cohomological type which are not lifted from smaller rank groups. These are conjecturally connected with reciprocity laws of Langlands type. Finally, the investigator and a colleague develop a theory of p-adic families of not necessarily p-ordinary cohomology classes for congruence groups. They use this to study the question whether non-lifted automorphic representations are "p-adically rigid". Algebraic number theory studies the properties of polynomial functions and equations whose coefficients are whole numbers. Ever since Descartes invented analytic geometry in the early 1600's, the ability to solve algebraic equations has been central to mathematics, science and engineering. In the last 50 years a new set of applications of number theory has opened up in the field of cryptography. The principal investigator studies delicate questions involving the fine structure and properties of solutions to certain systems of algebraic equations. Although one can often obtain approximate solutions using computers, there are many very subtle and difficult questions concerning the existence and form of the exact solutions. There are a number of conjectures on the table to explain what is going on, and the principal investigator has contributed some of them. He and his colleagues study these conjectures by verifying them in some cases by extensive computer calculations, and by proving them in special cases. At the core of this research is the surprising connection with geometric objects ("cohomology classes") that can be constructed in spaces of n-dimensional crystal lattices.