Galois Structure and Arithmetic Geometry

University Of Pennsylvania, Philadelphia PA

Investigators

Abstract

The principal investigator will apply tools from arithmetic geometry to study invariants attached to the actions of finite groups on schemes. These invariants are Euler characteristics associated to different kinds of cohomology. Methods of computing these Euler characteristics will be studied, along with their connection to L-series, Arakelov theory and the theory of motives. The principal investigator will also work on two other projects. These are to complete earlier research relating capacity theory to arithmetic intersections theory, and to study the universal deformation rings of representations of finite groups and their applications to Galois theory. This proposal concerns using geometric ideas to study the solutions of systems of algebraic equations. In geometry, it is often useful to consider the symmetries of an object, which are the ways the object can be rotated or flipped back onto itself. Another geometric idea, which goes back to the mathematician Euler, is that one can attach certain numbers ("Euler characteristics") to objects. These numbers provide a measure of complexity, and in simple cases they count various natural features, such as the number of holes in the object. The current proposal has to do with studying the symmetries of systems of equations, and how one can assign natural Euler characteristic numbers to them. The goal of this is to carry over to algebra some of the insights gained in geometry from considering symmetries and Euler characteristics. Systems of equations arise from many different mathematical applications. Euler numbers and symmetries can in many cases be used to either rule out or count the number of particular kinds of solutions to such equations.