MAT 482H1F - Lecture 101 (Fall 2024)

Composition laws, rings of low rank, and related counting problems

Prof. Ila Varma

Course Information

Lectures (online or hybrid): Tuesdays from 12pm-1pm on Zoom or MY 430
Lectures (online or hybrid): Wednesday from 12pm-2pm on Zoom or MY 430

First class is meeting on Tuesday, September 3rd, 2024.

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 3-torsion in class groups of quadratic fields. If interest and time permits, Bhargava's averaging methods to generalize results to quartic and quintic fields, applications of class field theory (if there is class interest), etc. 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%  
  • Problem Sets
 10%
  • Final Report
 20%
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. Discussions should include the following five areas:


The proposed punchline can be thought of as hints to help facilitators plan their discussion (where brackets [ ] denote a possible half-time goal).
After Tuesday, we should leave with: 

  • Problems for the problem set

  • Topics/Problems for Wednesday Lecture

  • Assigning Roles (latexing problem sets, doing problem sets, giving feedback to solutions)


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.


Problem Sets

Please participate in the problem sets, either by volunteering to latex problems from Tuesday discussions, doing problems on the communal set, or giving feedback to solutions (that have asked for feedback). Even if your solutions aren’t complete (i.e. you worked on a problem but didn’t get anywhere, but still added it to the communal problem set), you will get credit for submitting the work. 


Initial Survey & Final Report

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). The Final Report will consist of an at least one page latex document of what you learned in the course.


Initial Survey - Due Thursday, September 12th

(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 and the good fortune of the class assignment overlords, 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. 

 

The proposed punchline can be thought of as a possible goal for the Wednesday lecture (split by hour 1 and hour 2), and where brackets [ ] denote a possible half-hour goal.

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
 Proposed Punchline

1

 yes
 Overview of Lectures and Class Structure
2
 no
 Gauss Composition
 Section 3 of Seguin
Proposition 3.3
2
 Dirichlet Composition
  Section 4 of Seguin, Sections 3.1-3.2 of HCL1
 		Theorem 4.4-4.5
or Theorem 10
3
 no
 Composition via the Bhargava Cube
 Section 5 of Seguin, Sections 2.1-2.2 and Appendix of HCL1
 		after Theorem 5.6
or Theorem 1 + Appendix
3
 A first parametrization via the Bhargava Cube
 Section 2.3, Section 3.3 of HCL1
 		Theorem 11-12
4
Parametrizing with binary cubic forms, a modern view
 Section 2.4, Section 3.4 of HCL1
 		Corollary 14-15
4
 Parametrizing with binary cubic forms, the classical view  Levi-Delone-Faddeev Correspondence: Section 4 of GGS, Section 2.2 of HCL2, Section 2 of BST
 
Davenport-Heilbronn Correspondence: Section 6 of DH, Section 3 of BST		 Theorem 9 and 14
5
Rings and ideals parametrized by binary n-ic forms
 Sections 2.1-2.2 of Wood11
Section 1 of Nakagawa
 		Corollary 2.9
(or Corollary of Proposition 1.2)
5
 Parametrizations over cubic rings: 2 x 3 x 3 boxes of integers
 Section 1,  Section 2.1, Section 2.3 of HCL2
 		Theorem 2
6
Parametrizations over cubic rings, part 2: symmetrized boxes & resulting composition laws
 Section 2.4, Sections 3.1-3.2 of HCL2
 		[Theorem 4] Corollary 10
6
 Parametrization of ideal classes in rings associated to binary n-ic forms
 Wood14
(see also Section 2 of HSV) 		Theorem 1.3
(Theorem 2.2)
7
Resolvent rings
 Section 2 of HCL3
 		Equation 10
7
 Parametrization of quartic rings with cubic resolvents
 Sections 3.1-3.5 of HCL3
 		Theorem 1
8
 no
 Parametrization of quartic fields
 Sections 3.6-3.9 of HCL3
 		Corollary 5 and Corollary 18
8
 Quartic rings associated to binary quartic forms
 Wood12
 		Theorem 1.1
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
 		Equation 26
9
 The number of cubic rings of bounded discriminant, part 2: computing the volume
 Sections 5.4-5.5 of BST
 		Theorem 26
Week 10 - November 12th-13th, Ila is at a conference
11
 no
 Density of discriminants of cubic fields
 Lemma 19, Sections 8.2-8.4 of BST
 		Theorem 1 (or Theorem 8)
11
 Counting in the cusp
 Theorem 17, Section 4 of BV
 		Theorem 6
12
 yes
 Applications of class field theory to counting Section 8.1, Section 8.5 of BST, Section 5 of BV Theorem 2
12
 Density of discriminant of S4-quartic fields Bhargava05 Theorem 1 or 2
13
 no
 Final Reports Due