Modular forms

Final Paper

Topics have been chosen the week of April 4th. Please have a rough draft ready to submit to me by Thursday, April 29th (5pm). At this point it is not necessary to have the paper written in (La)TeX; a handwritten copy (double-spaced and with nice handwriting) is good enough (I have to be able to read it to make comments, and need room to write comments). Putting the paper into (La)TeX will take only a few days, perhaps longer if you're learning it for the first time (you'll thank me later on). The resources at the bottom of this page are very helpful introductions to (La)TeX; your more senior colleagues should also be helpful.

Overview

Modular forms play a crucial role in modern number theory. They are analytic objects that encode arithmetic data, and serve as a tool to investigate many phenomena (e.g. arithmetic of elliptic curves and Galois representations). This is a one-semester introduction to modular forms and their applications. The first half (or so) of the course will be devoted to basic material (the upper halfplane and the modular group, definition and first properties of modular forms, and Hecke and Atkin-Lehner theory); the second half will be an overview of different applications. For a more complete description you may refer to the course outline. For an introductory lecture about modular forms, you can consult the notes of my TWIGS talk.

Instructor

Prof. Paul Gunnells, LGRT 1115J, 545-6009, gunnells at math dot umass dot edu.

Office Hours

Tuesdays and Thursdays, 8:30-9:30, and by appointment.

Textbook

Elliptic Curves, Anthony Knapp, Princeton University Press. We won't follow this book, but it's a good reference for a lot of the basics and for perhaps the most important application of modular forms.

Each student will receive a copy of Don Zagier's Modular forms chapter in the From Number Theory to Physics (Les Houches, 1989) conference proceedings. This cannot be made available online, but I'll prepare copies by the second week of class or so.

Other handouts may be given during the term.

Additional resources can be found in the references section of the course outline. The following will be placed on reserve at the library:

• Knapp.
• Lang, Introduction to modular forms.
• Serre, A course in arithmetic.
• Shimura, Introduction to the arithmetic theory of automorphic functions.
• From Number Theory to Physics, Les Houches 1989.

Exercises

Occasionally during the course various results and examples will be assigned as exercises. These won't be collected and graded, but it will be expected that you think about them, write up arguments for your own purposes, and discuss them with me and your fellow students. In particular, anybody thinking they might want to go further in this or any related subject should take these very seriously!

Grading

The grades for this course will be based on a final paper. This will be an expository article of no less than five and no more than ten pages that you will prepare on an application of modular forms. My goal is to simulate as accurately as possible the experience of writing an original research paper. Here's how it will work:

• A list of subjects will be handed out when there are six weeks (or so) left in the term. I'll try to match topics to students based on their own interests (e.g. analysis, geometry, topology, number theory, etc.), and will give you pointers to the literature.
• You will research these topics outside of class and start to prepare a draft. You can ask me for advice if you need help (presumably you can also ask Prof. Hajir and Prof. Wong, but try not to bother them). As an example of what kind of writing I have in mind, you can consult the notes for my TWIGS talk. This paper is on the short side, since the talk was only an hour. You could pretend that you have two hours to give a lecture.
• A draft of your paper should be completed within three weeks (a deadline will be given later), which you will submit to me. The draft should be in some flavor of TeX (I use AMS-LaTeX); in particular this will be a good chance for you to learn TeX if you don't already know it. I will carefully read these drafts and make comments.
• You'll get the drafts back as soon as possible with comments. Then you will prepare a final draft that incorporates the requested changes, or whatever revisions are necessary.
• Then at the end of the term you'll submit the final version. These final versions will be made available to everyone in the class (barring any objections on the part of the individual authors).

Any kind of writing is challenging, and writing mathematics poses its own challenges. It's rare for a graduate student to get detailed feedback on writing before his or her thesis, but I feel that such feedback would have been extremely helpful to me. This should be a good opportunity for you to get some.

For more information about writing and mathematical writing in particular, you can consult the following:

• Mathematical writing, by Don Knuth (currently trying to get this on reserve at the library ... Mt. Holyoke has it).
• Milne's website. Take a look at the Tips for Authors link.
• J.-P. Serre's Hints on mathematical style. He's a master of mathematical exposition.
• A primer on using LaTeX can be found here.
• MIT has a writing requirement for its undergraduates, and Steve Kleiman has prepared an excellent introduction to writing short mathematical papers. There are also some notes available about using LaTeX, although a lot of features are peculiar to MIT's style files.
• Last but not least, there are several documents about TeX and LaTeX on the RCF page.

Additional stuff

• Check out the Modular Forms explorer, created and maintained by William Stein. This is a great way to poke around spaces of cusp forms of various weights and levels.
• This paper by Bill Casselman has a beautiful picture of the tessellation of the upper half plane by the SL_2(Z) translates of the fundamental domain. The paper's not about modular forms, but instead is about automatic structures in groups, an interesting subject in its own right. How he drew the picture (described in the text) is pretty amazing.
• Here you can find a program to produce pictures of fundamental domains of congruence subgroups.
• A mathematician in Switzerland commissioned a set of cabinets decorated with the tessellation of the upper half plane by translates of the fundamental domain. I wish I had enough money to get some for myself.