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 Grossencharacters 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 |