MAT 482H1F - Lecture 101 (Fall 2025)

Introduction to Arithmetic Invariant Theory & Arithmetic Statistics

Prof. Ila Varma

Course Information

Lectures (online or hybrid): Tuesdays from 12pm-1pm on Zoom or Bahen 2145
Lectures (online or hybrid): Wednesday from 12pm-2pm on Zoom or Myhal 317

First class is meeting on Tuesday, September 2nd, 2025.

Masks will be provided. Wearing masks is very strongly recommended. Please be mindful to those of us who are immunocompromised and have long-term chronic effects from COVID-19.

Instructor: Ila Varma (she/they)
 Office: Zoom (see Quercus for link or in person location)
Email: ila at math dot toronto dot edu 
(I prefer being contacted on Zulip)
 Office Hours: Tuesdays, 1-2pm


This course will offer students a way of learning or reviewing algebraic, analytic, arithmetic, and geometric material that pertains to the area within number theory known as “arithmetic statistics.” We will study composition laws, parametrizations of rings of low rank, and the foundational Davenport-Heilbronn Theorems on counting cubic fields or the average size of 3-torsion in class groups of quadratic fields. As most of this material is found in research articles rather than textbooks, much of the course will be dedicated to adjusting learning strategies to this format. .
References: Grading: 
  • Facilitating One Tuesday Discussion  
 25%  
  • Participation/Attendance in other Tuesday Discussions 
 25%  
  • Contributing to Lecture Notes/Book
 30%
20%

Please use Quercus to check grades and find lecture notes & recordings.
Please use Zulip for course discussion.

Facilitation

Each student will be responsible for facilitating Tuesday Discussion for one week during the semester. If there are more than 12 students, some students will pair up. An example agenda for a Tuesday Discussion is written below. The person facilitating the Tuesday Discussion can mold it to the material that week as you see fit.

After Tuesday, we should leave with:

  • Topics/Problems for Wednesday Lecture
  • Assigning Roles

Participation/Attendance in other Tuesday Discussions

Please attend and participate in Tuesday Discussions. Even if it is saying “I don’t understand,” or “I don’t know,” or even “I didn’t get a chance to do the reading,” it is important for the facilitator to have other active people there. Mathematicians rarely talk about mathematics they understand.

Lecture Notes/Book Contributions

Please contribute to the lecture notes, either by volunteering to latex lectures, background for lectures, proofreading, creating exercises, or producing "back of the book" solutions, or something else you can think of! Even if you are not totally sure if what you write is correct or complete or you mostly focus on correcting or contributing to other peoples’ writing, you will get credit for submitting the work. 


Initial Survey

The first assignment is a survey on course expectations, goals, and learning styles. It is required to turn in the first assignment on time (unless you contact the instructor).


Initial Survey - Due Friday, September 12th


Course Objectives


(Tentative) Schedule of Topics

This course will consist of a one hour meeting on Tuesday and a two hour meeting on Wednesday. Each Tuesday meeting (other than the first) there will be a student-led discussion of the material for that week. Together with the instructor, we will organize questions that arise in this discussion into problem set questions or questions to be answered during Wednesday's lecture.


This course will be mostly virtual, but with your help, it can be offered in a synchronous hybrid setting. In the first week, we will investigate the technological capabilities of the in person classrooms, and whether we can get volunteers to set up the classroom for a mutual viewing party in person, including offering masks and setting up the big screen/audio. Lectures will be recorded and available online asynchronously once they are completed. 

Assigning weeks to each student will happen in the first week. The course will also have a course discussion page on Zulip. 


Students are encouraged to work in groups to digest the material as well as to prepare for their facilitation. Students are also more than welcome to speak with me during office hours.
Week
 Is Ila in person?
 Title
 Reference Material

1

 yes
 Overview of Lectures and Class Structure
2
 no
 Gauss and Dirichlet Composition, classically

 Sections 3-5 of Seguin
 
3
 no
 Composition via the Bhargava Cube and a first parametrization
Sections 2.1-2.3 and Appendix of HCL1
 Section 3.1-3.3 of HCL1
4
Sept 23th-24th, no classes
5
 no
 Parametrizing with binary cubic forms, a modern view
 Section 2.4, Section 3.4 of HCL1
6
 no
 Parametrizing with binary cubic forms, the classical view Section 2 of BST or
  Section 2.2 of HCL2,
(optional reading: Section 4 of GGS, Delone-Faddeev, Levi)
7
 no
 Davenport-Heilbronn Correspondence, parametrizing maximal cubic rings
 Section 3 of BST,
(optional reading:
Section 6 of DH)
8
 no
 Local behavior and p-adic densities Section 4 of BST


Reading Week - Oct 29th-30th, no classes
9
 no
 The number of cubic rings of bounded discriminant, part 1: geometry of numbers Sections 5.1-5.3 of BST
10
 no
 The number of cubic rings of bounded discriminant, part 2: computing the volume
Sections 5.4-5.5 of BST
11
 no
 Density of discriminants of cubic fields
 Sections 8.2-8.4 of BST
12
Applications of class field theory to counting Section 8.1, Section 8.5 of BST
(optional reading: Section 5 of BV)