Modular Forms, Spring 2015


This is a Masters course in mathematics at the VU University Amsterdam.
Time and Place: Wednesday 10:00 - 12:45 in room P624 of the Mathematics and Science (W&N) Building.
First lecture: February 4
Teacher: Sander Dahmen, s.r.("last name")@vu.nl
Assistant: Abhijit Laskar, a.("last name")@vu.nl
ECTS Credits: 6
Form of tuition: Two hours of lecturing (with breaks), followed by a one-hour exercise session.
Examination: Homework exercises and a final oral exam on May 27.
Prerequisites: Basics of linear algebra, group theory, and complex analysis.
Aim: The aim of this course is to familiarize students with basic concepts, techniques, and applications of modular form theory as well as with some modern (deep) results about modular forms and their applications. The students will also learn how to perform explicit calculations with modular forms using the (free open-source) mathematics software system Sage.

Contents

This is an introductory course into the subject of modular forms and their applications. A modular form is a complex analytic function defined on the complex upper half plane which has a certain symmetry with respect to the action of SL(2,Z) (or some subgroup) on the upper half plane and which satisfies some growth condition. Near the turn of the 19th to 20th century it became clear that the coefficients of the series expansions are often integers with an important number theoretical interest. This insight has only grown in the 20th century and nowadays modular forms are an indispensable tool in modern number theory (but also play an important role in other subjects, e.g. physics). One of the great successes was their use in the proof of Fermat's Last Theorem by Andrew Wiles around 1994.
Topics include: the modular group and congruence subgroups, definition of modular forms and first properties, Eisenstein series, theta series, dimension of modular form spaces, Hecke operators, Petersson inner product. In the final part of the course we will give a global outlook on the use of modular forms and their associated Galois representations in the solution of Diophantine equations, in particular Fermat's Last Theorem.

Course material


For background reading and more we can also recommend


Program

References below are to the book by Diamond and Shurman, unless explicitly stated otherwise.
Exercises with a star (*) have to be handed in before/at the start of the next lecture.