Norm Varieties and Cohomological Invariants

Ohio State University Research Foundation -Do Not Use, Columbus OH

Investigators

Abstract

The investigator continues his work on the Bloch-Kato conjecture (bijectivity of the norm residue homomorphism) and on Galois-cohomological invariants for linear algebraic groups. Norm varieties form the basic examples of an intimate link between Galois cohomology and cobordism theory which lead in recent years to considerable progress on the Bloch-Kato conjecture. The main goal of the project is the construction of norm varieties for arbitrary weights and primes, the computation of some of their characteristic numbers and of some of their K-cohomology groups. Cohomological invariants are characteristic classes for G-torsors over a field. Classical examples are the discriminant and the Hasse-Witt invariant for quadratic forms. There is a huge variety of known cohomological invariants for various linear algebraic groups. One objective of this project is to get a better systematic understanding of cohomological invariants, but also to investigate certain particular cases. Some of the invariants (for the special linear group and for some exceptional groups) are closely related with norm varieties. A basic problem of algebra, number theory and algebraic geometry is to find solutions to polynomial equations. A mathematical "law" provides a means for solving a huge class of equations, rather than just finding a solution to a single equation. Thus the mathematician's goal is to find and verify useful mathematical laws. One potentially very powerful but as yet unverified law is called the "Bloch-Kato conjecture." With the goal of demonstrating that the Bloch-Kato conjecture is valid in general, the investigator studies in depth the properties of certain very special systems of equations, called "norm varieties."