Math 620, Algebraic Number Theory, 2014
Instructor:
Ching-Li Chai
Office: DRL 4N36, Ext. 8-8469.
Office Hours:
Email:chai@sas.upenn.edu
Notes and Homework Assignment
General Information
- Lectures: MW 12:00--1:30 PM, 3C8 DRL
First meeting: Wednesday, August 27, 2014.
- What is "basic algebraic number theory":
Stuffs in the following long ist are all considered "standard topics"
for a graduate number theory course.
- facts coming from local analysis:
local fields, splitting and ramification of primes,
including "higher ramification theory".
- global fields: adeles and ideles,
finiteness of class numbers, Dirichlet's unit theorem
- abelian zeta and L-functions: Dedekind zeta functions and Hecke L-functions,
functional equation of abelian L-functions.
These are most effectively treated using Tate's adelization method.
- Hecke L-functions (non-abelian)
- local class field theory
- global class field theory
- analytic methods involving L-funcitons: non-vanishing of L-functions,
distribution of primes, primes in arithmetic progression,
Brauer-Siegel theorem
- Reference Books:
There will be no texbook for this course. The following books are
recommended as references.
- Lang, Algebraic Number Theory
- Neukirch, Algebraic Number Theory
- Serre, Local Fields
- Cassel and Frohlich (eds.), Algebraic Number Theory
- Artin and Tate, Class Field Theory
- Davenport and Montgomery, Multiplicative Number Theory
- Washington, Cyclotomic Fields
- Hida, Elementary Theory of L-functions
- Weil, Basic Number Theory
Comments:
- Lang, Neukirch and Serre are good textbooks for standard
materials of algebraic number theory.
- Cassel/Frohlich is a
collection of notes for an instructional conference by many
authors, with a set of delicious problems by Tate at the end.
- Davenport/Montgomery is a good place to learn basic techniques of
analytic number theory.
- Hida's book is not elementary in the usual
sense; it is an excellent place to learn Iwasawa theoretic
approach to L-functions, after having absourbed some basics
from Washington's book.
- Weil's book is not as easy to read as
its title might suggest, but it will be a rewarding experience.
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