Preliminary Examination

Date: August 10, 2004; 2-4 pm
Examiners: Brian Conrad and Chris Skinner

Below is an official list of topics for my candidacy exam. Where applicable, I have included links to online resources as well as my personal notes.
The list of topics pdf tex (with the exception of the third topic, in my case abelian varieties), is standard for number theory Ph.D candidates.

Algebraic Number Theory

Basic algebraic number theory (local and global fields) Cassels-Frohlich
Milne's Math 676 notes
Stein's Math 129 notes
My notes on extensions of local fields, ramification groups etc. pdf
Local class field theory
Idelic and ideal-theoretic formulation of global class field theory
Ray class groups: definitions and descriptions in the above formulations
Kronecker-Weber Theorem
Chebotarev Density Theorem
Artin L-functions
Milne's Math 776 notes
The articles by Serre and Tate in Cassels-Frohlich pdf
My notes (based on the above two references) pdf
Neukirch: Algebraic Number Theory (esp. for Artin L functions)
Examples (quadratic fields, cyclotomic fields) Excercises from Cassels-Frohlich pdf
Z_p Extensions Washington: Cyclotomic Fields
Translation of Serre's "Classees des corps cyclotomiques" by Jay Pottharst

Algebraic Geometry

Basics of varieties
Basics of sheaves and schemes
Coherent cohomology of schemes
Curves (genus, Riemann-Roch,Hurwitz genus formula, Picard group, etc.)
Curves of genus zero (over any field)
Elliptic Curves (over any field)
Hartshorne, Chap. 2-4
My solutions to many of the exercises in the above (use at your own risk) pdf
Tate's article "The arithmetic of elliptic curves"

Abelian Varieties

Basics of Abelian Varieties
Isogeny-invariance of BSD
Mumford: Abelian Varieties
Milne's article in "Arithmetic Geometry"
Milne's Math 731 notes
Milne's "Arithmetic Duality Theorems" (esp. Section 7)
My notes based on ADT pdf