Modular Forms - 8EC

Prerequisites

Basic knowledge of group theory (groups, (normal) subgroups, cosets, quotients, first isomorphism theorem, group actions) and complex analysis (holomorphic/meromorphic functions, Taylor/Laurent series, path integrals, the residue theorem). These concepts are covered by most introductory courses in group theory and complex analysis, respectively, and can be found in almost any standard text on (abstract) algebra and complex analysis, respectively (see e.g. texts by Ahlfors or Lang).

Aim of the course

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 SageMath.

Rules about Homework / Exam

The final mark will for 25% be based on regular hand-in exercises and for 75% on a final written exam, with the extra rule that in order to pass the course the student needs to score at least a 5.0 on the final exam.

Lecture Notes / Literature

Lecture notes will be made available before the start of the course. For background reading and more we can also recommend:

• F. Diamond and J. Shurman, "A First Course in Modular Forms", Graduate Texts in Mathematics 228, Springer-Verlag, 2005. (Covers most of what we will do and much more.)
• J.-P. Serre, "A Course in Arithmetic", Graduate Texts in Mathematics 7, Springer-Verlag, 1973. (Chapter VII is very good introductory reading.)
• W.A. Stein, "Modular Forms, a Computational Approach", Graduate Studies in Mathematics, American Mathematical Society, 2007. (Emphasis on computations using the free open-source mathematics software system SageMath.) See also the L-functions and Modular Forms Database <http://www.lmfdb.org/>.
• J.S. Milne, "Modular Functions and Modular Forms", online course notes <http://www.jmilne.org/math/CourseNotes/mf.html>.
• J.H. Bruinier, G. van der Geer, G. Harder and D. Zagier, "The 1-2-3 of Modular Forms", Universitext, Springer-Verlag, 2008.

Lecturers

S. Dahmen (VU) & P. Bruin (UL)