Local Global Principles for Galois Cohomology

Joint IAS/PU Number Theory Seminar
Topic:Local Global Principles for Galois Cohomology
Speaker:Julia Hartmann
Affiliation:RWTH Aachen University; Member, School of Mathematics
Date:Thursday, December 13
Time/Room:4:30pm - 5:30pm/S-101
Video Link:https://video.ias.edu/1213/members/1213-JuliaHartmann

We consider Galois cohomology groups over function fields F of curves that are defined over a complete discretely valued field. Motivated by work of Kato and others for n=3, we show that local-global principles hold for H^n(F, Z/mZ(n-1)) for all n>1. In the case n=1, a local-global principle need not hold. Instead, we will see that the obstruction to a local-global principle for H^1(F,G), a Tate-Shafarevich set, can be described explicitly for many (not necessarily abelian) linear algebraic groups G. Concrete applications of the results include central simple algebras and Albert algebras.