Course Notes
Group Theory
Fields and Galois Theory
Algebraic Geometry
Algebraic Number Theory
Modular Functions and Modular Forms
Elliptic Curves -- see books.
Abelian Varieties
Lectures on Etale Cohomology
Class Field Theory
Algebraic Groups, Lie Groups, and their Arithmetic Subgroups
Complex Multiplication
pdf file for the current version (3.00)
This work is a modern exposition of the theory of algebraic groups (affine group schemes), Lie algebras, Lie groups, and their arithmetic subgroups.

Contents

I: Basic Theory of Algebraic Groups Definition; examples; some basic constructions; affine groups and Hopf algebras; affine groups and affine group schemes; group theory: subgroups and quotient groups; representations of affine groups; group theory: the isomorphism theorems; recovering a group from its representations; Jordan decompositions; dharacterizations of categories of representations; finite flat affine groups; the connected components of an algebraic group; groups of multiplicative type; tori; unipotent affine groups; solvable affine groups; the structure of algebraic groups; the spin groups; the classical semisimple groups; the exceptional semisimple groups; tannakian categories.
II: Lie Algebras and Algebraic Groups The Lie algebra of an algebraic group; Lie algebras and algebraic groups; nilpotent and solvable Lie algebras; unipotent algebraic groups and nilpotent Lie algebras; semisimple Lie algebras and algebraic groups; semisimplicity of representations
III: The Structure of Semisimple Lie Algebras and Algebraic Groups in Characteristic Zero Root systems and their classification; structure of semisimple Lie algebras and their representations; structure of semisimple algebraic groups and their representations; real Lie algebras and real algebraic groups; reductive groups.
IV: Lie groups Lie groups; Lie groups and algebraic groups; compact topological groups
V: The Structure of Reductive Groups: the split case Split reductive groups: the program; the root datum of a split reductive group; Borel fixed point theorem and applications; parabolic subgroups and roots; root data and their classification; construction of split reductive groups: the existence theorem; Construction of isogenies of split reductive groups: the isogeny theorem; representations of split reductive groups.
VI: The Structure of Reductive Groups: general case The cohomology of algebraic groups: applications; Classical groups and algebras with involution; relative root systems and the anisotropic kernel
VII: Arithmetic groups

History

v0.00 (May 22, 2005; 219 pages). Posted as Algebraic Groups and Arithmetic Groups (AAG).
v1.00 (April 27, 2009; 192 pages). First version of expanded notes (first two chapters only).
v2.00 (April 27, 2010; 378 pages). Posted new version of all six chapters.
v3.00 (April 1, 2011; 422 pages). Major revision (now seven chapters).