## matthias beck

some of matt's pictures
 MATH 890 Advanced Number Theory: Modular Forms Spring 2014

 Lecture Tue/Thu 9:35-10:50 BUS 128 Instructor Dr. Matthias Beck Office Thornton Hall 933 Office hours Tue 2:00-3:30, Wed 11:00-12:00, Thu 3:30-5:00 & by appointment

Course objectives. Modular forms are meromorphic functions on the upper half plane that satisfy fundamental symmetry and growth conditions. Roughly speaking, you can think about them as functions that are invariant under Möbius transformations. (The Escher picture above essentially illustrates this invariance, after the upper half plane gets conformally mapped to the unit disk.) Modular forms are essential tools for the study of many elementary number-theoretic functions, such as the partition function (how many ways are there to write n as a sum of integers?), the sum-of-divisors function (sumd|n da for some fixed a in R), and the sum-of-squares function (how many ways are there to write n as a sum of k squares?), they form a powerful connection between algebraic and analytic number theory (as exemplified by Wiles' proof of Fermat's Last Theorem), but they also have a beautiful life of their own. We will study classical aspects of modular forms and some of their applications to elementary (but hard) problems in Number Theory.

Prerequisites. Undergraduate Complex Analysis. This prerequisite will be waived for students who are taking Math 730 concurrently with my course.

Text book: Tom M. Apostol, Modular Functions and Dirichlet Series in Number Theory, 2nd edition, Springer.

Syllabus: I plan to cover Chapters 1, 2, 3, and 5 in the text book, that is:

• Elliptic functions, Weierstrass p-function, Eisenstein series, discriminant function Delta, Klein's modular function J, Fourier expansions
• Möbius transformations, modular group, fundamental regions, modular functions, Picard's Theorem
• Dedekind eta-function and functional equation, Dedekind sum and its reciprocity law
• The circle method and Rademacher's series for the partition function

Grading system: I will assign weekly homework problems, due on Thursdays before class (90% of the final grade). In addition, every student will give one lecture on the material we will cover (10% of the final grade; here is a list of possible topics).

I want to ensure that each of you accomplishes the goals of this course as comfortably and successfully as possible. At any time you feel overwhelmed or lost, please come and talk with me.

This syllabus is subject to change. All assignments, as well as other announcements on tests, policies, etc., are given in class. If you miss a class, it is your responsibility to find out what's going on. I will try to keep this course web page as updated as possible, however, the most recent information will always be given in class. Always ask lots of questions in class; my courses are interactive. You are always encouraged to see me in my office.