## Math 9800, Fall 2020

### Topic Outline:

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 that 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), have led to a flurry of activity just in the past few years. 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.

### Rough schedule:

DateTopics coveredNotes
August 24 Introduction Lecture 1 Notes
August 26 Motivating Exampls of Modular Forms Lecture 1 Notes
August 28 Elliptic Functions: Definitions and basic facts Lecture 2 Notes
August 31 Elliptic Functions: Constructions and the Weierstrass p-function Lecture 3 Notes
September 2 The field of elliptic functions and a special differential equation Lecture 4 Notes
September 4 Elliptic Curves Lecture 5 Notes
September 7 More on Elliptic Curves Lecture 6 Notes
September 9 Connections with Ellitpic Functions and the Congruent Number Problem Lecture 7 Notes
September 11 Torsion points on the congruent number elliptic curves Lecture 8 Notes
September 14 Modular Forms: The basics Lecture 9 Notes
September 16 The fundamental domain Lecture 10 Notes
September 18 Eisenstein Series Lecture 11 Notes
September 21 The Valence Formula Lecture 12 Notes
September 23 Dimensions of modular form spaces Lecture 12 Notes
September 25 Modular functions and elliptic curves Lecture 13 Notes
September 28 Differential operators and quasimodular forms Lecture 14 Notes
September 30 The Ramanujan tau function Lecture 15 Notes
October 2The Dedekind eta function Lecture 16 Notes
October 5Partition functions and Tauberian theorems Lecture 17 Notes
October 7Hecke operators Lecture 18 Notes
October 9New modualr forms from old ones Lecture 18 Notes
October 12More on Hecke operators Lecture 19 Notes
October 14Bases of eigenforms and the Petersson inner product Lecture 20 Notes
October 16 L-functions and their analytic continuation and functional equations Lecture 21 Notes
October 19Theta functions, Poisson summation Lecture 22 Notes
October 21Sums of square identities, generalized theta functions Lecture 22 Notes
October 23Generalized theta functions Lecture 22 Notes
October 26 Weil Conjectures and the zeta function of the congruent number curves Lecture 23 Notes
October 28Gauss and Jacobi sums and the zeta function of the congruent number curves Lecture 23 Notes
October 30L-functions of elliptic curves, and connections with modular forms (Modularity Theorem/Birch and Swinnerton-Dyer Conjecture) Lecture 24 Notes
November 2Atkin-Lehner-Li Theory, Eichler-Shimura theory, and motivating ideas behind the proof of Fermat's Last Theorem Lecture 25 Notes
November 4Motivating ideas behind the proof of Fermat's Last Theorem Lecture 25 Notes
November 6Applications to the congruent number problem Lecture 26 Notes
November 9Class Cancelled
November 11Poincaré series of different types Lecture 27 Notes
November 13The Shimura/Shintani correspondence and Tunnell's Theorem on congruent numbers Lecture 27 Notes
November 16Complex Multiplication Lecture 28 Notes
November 18Singular Moduli Lecture 28 Notes
November 20Class polynomials Lecture 28 Notes
November 30Class number relations and Kronecker's Jugendtraum Lecture 28 Notes
December 2Moonshine Lecture 29 Notes
December 4Sphere packing Lecture 30 Notes