Course 18.784: Seminar in Number Theory

Final papers

Final papers are due on Wednesday, May 12, in class (or earlier). This deadline is essentially absolute, barring serious emergencies (i.e., involving notice from the appropriate office).

Before the papers are due, I can look over drafts and give a rough prioritized list of problems. I am also free to discuss questions at length during scheduled office hours and by appointment.

Here are some absolute requirements - breaking these is a good way to fail:

• There is an approximate lower bound on length: 10 pages. I strongly recommend against using tricks with margins, font size, etc. to get past this bound. If you find yourself running out of things to say, I suggest you pick a simple but nontrivial example computation that is on topic, and insert it in an appropriate place with some context.
• You must have a bibliography, and you must cite work and resources that you used for your paper.
• Your paper must be in a form that I can read. Not allowed: electronic submissions in file formats like Word(tm), non-quoted text in non-English languages (even if you have a good reason to think I can read your language of choice), diagrams in crayon.

Here is a rough sketch of grading considerations (still in flux):

• Exposition - I'd like to understand the point of each paragraph. Ideas should flow reasonably naturally. There should be an introduction and a conclusion.
• Technical content - The mathematics should be correct and on topic. Derivations should be motivated.

Project Topics

• MD, Elliptic functions
• JE, The Kodaira-Spencer style isomorphism
• PE, Orbifold Riemann-Roch on modular curves
• AF, Siegel modular forms
• AH, Atkin-Lehner involutions and genera of X₀(N)⁺
• CK, Genera of modular curves
• SK, Arithmetic hyperbolic 3-manifolds
• WS, Characters of integrable highest-weight representations of affine Lie algebras
• BW, The circle method of Hardy-Littlewood-Rademacher
• QY, Computational Eichler-Shimura
• AZ, Class groups and integrality of j

Topic suggestions

I'm collecting some ideas for projects here. Some of them are rather broad, and you can make a fine project out of a small part of a general topic. Multiple students can work on a single topic, as long as they are studying different facets of the subject. I think it is important that you find a topic that interests you and is at a suitable level. There is no extra credit for making your life miserable!

This list is in a preliminary form. I'll continue to refine it.

Analysis

The Hardy-Littlewood-Rademacher circle method

This is a clever contour integral technique for computing the q-expansion of a negative weight modular form, using only the singular part of the expansion. You end up with a series of modified Bessel functions that converges spectacularly quickly. For example, the q¹⁰ coefficient of 1/Δ is 2705114880, and the first term in the series is about 2705114880.565. Since the coefficents in this case are integers, you only need the first two terms to get the right answer.

Real-analytic forms

These are functions on the upper half plane that are not holomorphic, but satisfy differential equations that are similar to Cauchy-Riemann. Examples include real-analytic Eisenstein series.

Discrete series representations of SL₂(R)

This may be a bit advanced, but if you really like Hilbert space, it could be rewarding.

Group theory

Bruhat-Tits trees and Helling's theorem

A Bruhat-Tits tree can be viewed as a combinatorial analogue of the upper half plane. There is a very rich theory surrounding these objects and their higher-dimensional incarnations (called buildings), but they also provide a simple way to enumerate certain discrete subgroups of SL₂(R) that are about the same size as SL₂(Z).

Genera of modular curves

I am imagining this as a combination of theory and computer work. The quotient of the upper half plane by a finite index subgroup of SL₂(Z) is a Riemann surface with finitely many punctures. Learn to work out formulas for the number of handles and punctures for various groups.

Algebraic geometry

Orbifold Riemann-Roch

This is a fancy way to compute the spaces of modular forms on a given quotient of the upper half plane. You should be pretty comfortable with line bundles.

The Kodaira-Spencer style isomorphism

There is another way to view modular forms, using differential forms on families of elliptic curves. This project will be an introduction to elementary deformation theory.

Computational Eichler-Shimura

The actual Eichler-Shimura correspondence is rather subtle, but you can gather evidence for it with explicit computations. This is a way to learn about Jacobians, Hecke operators, Galois theory, and mod p geometry.

Class fields and special functions

There are a few potential projects here. Perhaps the most famous concerns the fact that exp(π√163) is very close to an integer, and its explanation in terms of properties of the modular j function. Another concerns the construction of abelian extensions of imaginary quadratic fields via elliptic functions.

Generalizations of modular forms

Theta functions as Jacobi forms

The theory of Jacobi forms is roughly what you get when you cross the theory of elliptic functions with the theory of modular forms.

Modular forms of half-integer weight

Natural examples of these objects arise from theta functions of odd-dimensional lattices, and eta-products. One can compute interesting examples, and dimensions of spaces of them.

Vector-valued forms

Given a finite dimensional representation of SL₂(Z), one can consider forms that take values in a complex vector space of dimension greater than one, and transform in a manner compatible with the representation. This is one way to subsume the theory of modular forms for subgroups of SL₂(Z).

Hilbert and Siegel forms

These are versions of modular forms on spaces of real dimension greater than two. A Hilbert modular form lives on a product of half planes, and transforms well with respect to a group over the ring of integers of a totally real field (instead of just Q). A Siegel modular form lives on a space parametrizing abelian varieties, which are a higher dimensional generalization of elliptic curve.