Lecture 14: Galois Cohomology of Abelian Varieties
over Finite Fields
William Stein
Feb 12, 2010
See also Pete Clarks http:/math.uga.edu/~pete/wcnotes.pdf.
1
Principal Homogenous Spaces for Abelian Varieties
An abelian variety A over a eld k is a projecti
Lecture 13: Brauer Groups
William Stein
Feb 10, 2010
This lecture is about Brauer groups.
Reference: Chapter X of Serres Local Fields.
1
The Denition
Let k be a eld, and x a separable closure k sep of k .
Denition 1.1. The Brauer group of k is
Brk = H2 (k
Lecture 12: Kummer Theory
William Stein
Feb 8, 2010
1
Kummer Theory of Fields
Kummer theory is concerned with classifying the abelian extensions of exponent n of a
eld K , assuming that K contains the nth roots of unity. Its a generalization of the
corres
Lecture 11: Galois Cohomology
William Stein
Feb 5, 2010
1
Galois Cohomology
In this course we have developed a foundation for group cohomology. The goal for the
rest of the course (about 15 lectures), is to see some applications of group cohomology
to Gal
Lecture 10: Cup Product
William Stein
Jan 27, 2010
1
Cup Product
1.1
Introduction
We will dene and construct the cup product pairing on Tate cohomology groups and
describe some of its basic properties. The main references are 7 of Atiyah-Wall, VIII.3
of S
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