MAT1210HF & Fields Academy: Class Field Theory (Fall 2024)

Prof. Ila Varma

Course Information

Lectures (online or hybrid): Tuesday from 2pm-4pm          (Classroom Location at UofT to be determined)

Lectures (online or hybrid): Wednesday from 3pm-4pm     (Classroom Location at UofT to be determined)

This course will be delivered online, through Zoom.

First class meeting is on Tuesday, September 10th, 2024 on Zoom.
All lectures will be recorded and made available through the Class Zulip or this website.

Instructor: Ila Varma (she/they)
 		Office: Class Zoom Link
Email: ila at math dot toronto dot edu
(but I prefer being contacted on Zulip)
 Office Hours: Wednesdays 2pm-3pm
.
   Course Textbooks:

Additional References:

 
Prerequisites: MAT 415/1200 or one of the following texts:

References for Algebra Background


 
Problems and Grading

Grades will be based on participation. The first assessment of the course will request each student to describe their learning style, how they want to participate in the course, and other information for smooth running of the course. It is optional to provide me with any of this information, but it is mandatory to fill out the form.

Auditing will be allowed, but you still must fill out the form for Home Assessment 1.

You are encouraged to work together on problems.


Home Assessment 1 (due 9/20/24)

  
Schedule of Lectures

In the below schedule, [S] refers to Sutherland's notes
Week #
 Lecture Dates
 Material covered
 References for material covered
1
 9/10/24
9/11/24
 		Intro, Review of Galois Theory, Galois Theory and Prime Ideals, Decomposition and Inertia Groups, Frobenius elements Section 24.3 of [S],
Section 21.2 of [S],
Section 13.1 of [S],
Section 5.3 of [S],
Section 7.1-2 of [S]
2
 9/17/23
9/18/24
HW Due 9/20/24
 		Decomposition and Inertia Groups (cont'd), Frobenius (cont'd), CFT Rebrand, Introduction to Local Fields, Galois theory of Local Fields
Section 7.2-3 of [S],
Chapter 8 of [Marcus],
Section 8.2 of [S],
Section 8.1 of [S],
Section 11.3-4 of [S]
3
 9/24/24
9/25/24
 		Connecting Local and Global Fields, Local Kronecker-Weber implies Global Kronecker-Weber, Fractional Ideals, The Importance of Splitting Completely, Introducing the Artin Map
 Section 11.3-4 of [S],
Section 20.1 of [S],
Section 2.5 of [S],
Section 7.4 of [S]
Section 21.1 of [S]
4
 10/1/24
10/2/24
 		Introducing the Artin Map (cont'd), Global CFT statement and example utilizing the Artin Map, Abstracting our previous take of Local Fields, Hensel's Lemma: statement, applications and proof
Section 21.1 of [S]
Section 7.4 of [S],
Section 8.1-2 of [S],
Section 9.1 of [S],
Section 1.2 of [S],
Section 1.4 of [S],
Section 9.2 of [S]
5
 10/8/24
10/9/24
Hensel's Lemma (proof cont'd), proving local Kronecker-Weber, Re-introducing CFT with ray class language 
Section 9.2 of [S]
Section 20 of [S]
lmfdb.org
Section 21.2 of [S]
6
 10/15/24
10/16/24
 		Introduction/Review of places, Introduction to modulus, ray class groups, ray class fields, Weak Approximation, ray class number formula and exact sequence, Dedekind zeta functions Section 1.2 of [S]
Section 13.1 of [S]
Section 9.1 of [S]
Section 21.3 of [S]
7 10/22/24
10/23/24
 Dedekind zeta functions (continued), polar density, The Importance of (Sets of Primes) Splitting Completely (cont'd), conductors, norm groups
 Section 21.4-5 of [S]
Section 22.3 of [S]

Fall Break No Classes
8
 11/5/24
11/6/24
 		Norm groups, congruence groups, ideal-theoretic CFT (precise version), Fundamental Inequalities of CFT (towards the proof of Artin Reciprocity) 
Section 22.4-5 of [S]
Section 24.2 of [S]
9
 11/12/24
11/13/24
 		Group cohomology, cohomology via free resolutions, Homology, tensors, Tate cohomology of cyclic groups
 Section 23.1-2 of [S]
Section 23.4-5 of [S]
10
 11/19/24
11/20/24
 		Applications of Tate Cohomology of Galois groups of global field extensions, Hilbert Theorem 90, Herbrand Unit Theorem, Proof of the isomorphism of Artin Reciprocity in the cyclic unramified case, Ambiguous Class Number Formula
 Section 24.1-2 of [S]
11
 11/26/24
11/27/24
 		Generalizing to noncyclic unramified case of Artin Reciprocity, Adeles, Ideles, idele class group, strategy for extending to the ramified case
 Section 24.3-4 of [S]
Section 25.3-4 of [S]
12 12/3/24
12/4/24
 		ideles continued, norm groups, idelic local and global class field theory statements,
Section 27-27.3 of [S]
Section 28-28.3 of [S]



Accommodations

The University provides academic accommodations for students with disabilities in accordance with the terms of the Ontario Human Rights Code. This occurs through a collaborative process that acknowledges a collective obligation to develop an accessible learning environment that both meets the needs of students and preserves the essential academic requirements of the University's courses and programs.

Students with diverse learning styles and needs are welcome in this course. If you have a disability that may require accommodations, please feel free to approach Ila. If it is more comfortable for you or for any other class you are in, please approach the Accessibility Services* office.

On Respectful Learning

All members of the learning environment in this course should strive to create an atmosphere of mutual respect where all members of our community can express themselves, engage with each other, and respect one another's differences.

Recording of Online Course Meetings
This course, including your participation, will be recorded on video and will be available to students in the course for viewing remotely and after each session.

Course videos and materials belong to your instructor, the University, and/or other sources depending on the specific facts of each situation and are protected by copyright. In this course, you are permitted to download session videos and materials for your own academic use, but you should not copy, share, or use them for any other purpose without the explicit permission of the instructor.

For questions about the recording and use of videos in which you appear, please contact the instructor.

Technology

Lectures can be accessed through Zoom, either synchronously or asynchronously. In addition, home assessments will be turned in via upload on Crowdmark. Please see the following links for the general technological requirements needed for the course.

If you do not have access to such technology, please contact the instructor.

Missed Assignments
 
A verification of illness (also known as a "doctor's note") is temporarily not required. Students who are absent from class for any reason (e.g., COVID, cold, flu and other illness or injury, family situation) and who require consideration for missed academic work should report their absence through the online absence declaration. The declaration is available on ACORN under the Profile and Settings menu. Students do not need to feel obligated but are welcome to advise Ila of their absence. Visit COVID-19 Information for University of Toronto Students page on the Vice-Provost, which includes information on this and other frequently asked questions.