Instructor: Ching-Li Chai

Office: DRL 4N36, Ext. 8-8469.
Office Hours: M 1:30-2:30 pm, and by appointments.

General Information

• Masks are required

• Lectures:
  Spring 2023, MW 12:00--1:29 PM, 4E9 DRL
  First meeting: Wednesday, January 11, 2023.
  
  Fall 2022, MW 12:00--1:29 PM, 3N6 DRL
  First meeting: Wednesday, August 31, 2022.
  
  
• What is "basic number theory": Stuffs in the following long list 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
  
  
  
• Plan for the fall semester 2022: We will only be able to cover some of the above topics.
  1. Elementary topics: algebraic integers, completions, ramification theory (different, discriminant and higher ramification theory). Most of these can be found in chapters 1-4 of Lang's Algebraic Number Theory, and also chapters 1-4 of Serre's Local Fields. We will do these quickly, say three weeks. The higher ramification can be a topic some of you present in class.
  2. Definition and basic properties of adels and ideles, ending with the adelic version of Dirichlet's theorm on finiteness of class numbers. (The statement is that norm one subgroup of the idele class group attached to an algebraic number field is compact.) Here we will have to use some analysis, including the exsistence of Haar measures on locally compact groups, and some basic properties of Fourier analysis on locally compact abelian groups. We will give the full statements of these properties; you can assume/admit these and proceed with confidence, or you can look up your preferred reference. It takes some time to get used to thinking about things adelically, but the effort is well worth it.
  3. Basic properties of zeta and L-functions. We will introduce the approach in Tate's thesis: define local zeta functions and determine their functional equations by elementary calculations, then putting the local functional equations together using the Poisson summation formula.
  4. A quick introduction to class field theory. We will not have time for the proofs. Instead we will give the statements and illustrate them in examples.
  
  
  
• Background knowledge: basic algebras and analysis at the level of standard first year graduate courses, specified below.
  • Noetherian commutative rings, prime ideals, localization of commutative rings and modules over commutative rings, Nakayama's lemma, integral closure of a integral domain, discrete valuation rings, Dedekind domains, Galois theory for finite extensions of fields
  • basic measure theory and integration
  • holomorphic and meromorphic functions, residue calculus, basic Fourier series and integrals, Poisson summation formula
  
  
  
• 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.
  Important Dates:
  
  • First meeting of classes, Spring 2023: Wednesday, January 11
    
  • MLK Jr. Day: Monday, January 16
    
  • Drop period ends: Monday, February 20
    
  • Spring break: March 4 (Saturday)-12 (Sunday)
    
  • Last day to withdraw: Monday, March 27
    
  • Last day of classes: Wednesday, April 26
    
    
  • First meeting of classes, Fall 2022: Tuesday, August 30
    
  • Labor Day (no class): Monday September 5
    
  • Drop period ends: Monday October 10
    
  • Last day to withdraw: Monday November 7
    
  • Thanksgiving break: November 24-27
    
  • Last day of classes: Monday December 12
    
  • Reading days: Tuesday December 13-Wednesday December 14
    
  
  

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  Last modified: Tue Jan 10 21:37:54 EST 2023