Special Topics course offered in Spring 2018 by Larry Rolen on "Modular Forms: Theory and Applications"

N. Koblitz: Elliptic curves and modular forms

T. Apostol: Modular Functions and Dirichlet series in number theory

According to Martin Eichler, there are five fundamental operations of mathematics: addition, subtraction, multiplication, division, and modular forms. In this course, students will learn how to justify this claim, and in particular will see many of the beautiful applications of modular forms to number theory and other areas of mathematics. For example, modular forms are central to the proof of Fermat's Last Theorem, and can be used to show other Diophantine results, such as the fact that 144 is the largest Fibonacci number which is also a perfect power. Modular forms have a knack for showing up in surprisingly deep proofs of very simple-to-state results like these, and of many surprising facts, such as the seemingly innocuous (but very deep) observation e^{pi*sqrt(163)) 262537412640769743.99999999999925 is incredibly close to being an integer. These applications continue to arise in hot-topic areas of mathematics; in fact, modular forms proofs of cases of the sphere packing problem, which asks for optimal arrangements of spheres to fill up as much space as possible (think stacking of oranges in a grocery store), were published in 2017.

Roughly speaking, modular forms are complex functions which are periodic, like sine or cosine, but satisfy infinitely many more symmetries simultaneously. This may seem surprising at first, and satisfying infinitely many symmetry properties is indeed very constraining. In fact, it allows one to use basic complex analysis to build up a very rigid algebraic theory of these functions. This is also what makes modular forms so special and where applications to arithmetic and number theory arise; for example, these symmetries turn them into a tool for proving infinitely many identities with a finite computer check.

In this course, we will survey this theory and its applications, as well as its connections to other objects of number theory such as elliptic curves and elliptic functions, with an eye towards understanding Tunnell's criterion determining which integers n are congruent (that is, areas of right triangles with rational side lengths... try to see if you are able to determine a few examples for youself!).

Prerequisites: No knowledge of elementary number theory is assumed, however it would be recommended that students are familiar with applying the main theorems of complex analysis (Identity theorem, Cauchy's theorem, Residue theorem, etc.). However, the basic uses of these theorems can also be learned concurrently, depending on the background of the class.