Cohomology of locally symmetric spaces and applications to number theory

Duke University, Durham NC

Investigators

Abstract

The cohomology of locally symmetric varieties plays an important role in number theory and in particular Langlands's program. One reason is that they are examples of spaces exhibiting an action of both a Hecke algebra] and a Galois group. A fundamental goal is to relate these actions and the corresponding L-functions. In previous work, the principal investigator created a new tool, L-modules, to study such cohomology. In the current project, the principal investigator proposes to incorporate Hecke and Galois actions into L-modules. He will also prove a "decomposition theorem" for L-modules and apply it to construct cycles associated to sub-locally symmetric varieties. The proposed research studies symmetry in geometry and number theory. Applications of geometry and number theory abound, for example to cryptography and crystallography. A symmetry in geometry is a transformation of space which doesn't change distance, for example a rotation. In number theory a symmetry is a transformation of a number system that transforms the sum of two numbers into the sum of the transforms and likewise for the product, for example complex conjugation. In both subjects, the most interesting objects are those which remain unchanged under many symmetries. For example, in geometry the sphere is invariant under all rotations, while in number theory the set of roots of a polynomial equation with integer coefficients are invariant under all symmetries. The investigator's research concern higher-dimensional objects which possess both geometric and number theoretic symmetries, and the relations between these symmetries.