MAT1210HF: Class Field Theory (Fall 2023)

Prof. Ila Varma

Course Information

Lectures (online or hybrid): Tuesday from 1pm-3pm          (Classroom Location: AP 124 at 19 Ursula Franklin Street)

Lectures (online or hybrid): Wednesday from 12pm-1pm     (Classroom Location: MS 3278 at 1 King's College Circle)

This course will be delivered online, through Zoom. (Please see Quercus for the Zoom Link).
For most weeks, I will ALSO offer the option to watch and participate in person. It is required to wear MASKS!
Please pay attention to announcements on Quercus and/or the calendar below for updates regarding course delivery.

First class meeting is on Tuesday, September 12th, 2023 on Zoom or in BL 112 (Claude T. Bissell Building at 140 St. George Street).
Please join there or virtually through the Zoom link provided on Quercus. All lectures will be recorded and made available on Quercus.

Instructor: Ila Varma (she/they)
 		Office: Bahen Building, Room 6108 and/or Class Zoom Link
Email: ila at math dot toronto dot edu
(but I prefer being contacted on Zulip)
 Office Hours: Wednesdays 1pm-2pm
.
 You can find an invitation to the course discussion page on Zulip also through Quercus.

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.

You are encouraged to work together on problems.


Home Assessment 1 (due 9/21/23)

  
Schedule of Lectures

In the below schedule, [S] refers to Sutherland's notes
Week #
 Lecture Dates
 Will Ila be in person or just online?
 Material covered
 References for material covered
1
 9/12/23
9/13/23
 		Hybrid/In Person
 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
 9/19/23
9/20/23
HW Due 9/21/23
 		Online 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
 9/26/23
9/27/23
 		Online 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
 10/3/23
10/4/23
 		Online 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
 10/10/23
10/11/23
Online 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
 10/17/23
10/18/23
 		Online 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 10/24/23
10/25/23
 Hybrid/In Person 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]
8
 10/31/23
11/1/23
 		Online 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]

Fall Break

No Classes
9
 11/14/23
11/15/23
 		Hybrid/In Person TBD
10
 11/21/23
11/22/23
 		Online TBD
11
 11/28/23
11/29/23
 		TBD TBD
12 12/5/23
12/6/23
 		TBD TBD

2023/24 Junior Number Theory Seminar - Wednesdays at 3pm
This seminar discusses modern number theory while our class is focused on early 20th century so there is a bit of a gap between the material in this course and the seminar. However, it's a great place to see what kinds of number theory are interesting to your fellow graduate students and build community!

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.