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. The above also applies to the retake.
There will be 6 sets of homework exercises. The average mark of the best 5 of those sets counts for 25% towards the final grade.
Homework exercises have to be handed in online, as pdf file. Please use the links at the bottom of the "Homework Exercises" section.
Solutions made with LaTeX are preferred, but we will also accept scans/pictures of written homework; in that case please make sure these are of good quality.
Cooperation is allowed (and even encouraged), but every student has to hand in their own version in their own words. Copying (parts of) homework is considered fraud and can have severe consequences.
Lecture notes/Literature
Lecture notes will be uploaded here before the start of the course.
Some additional material will be made available during 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.H. Bruinier, G. van der Geer, G. Harder and D. Zagier, "The 1-2-3 of Modular Forms", Universitext, Springer-Verlag, 2008.
- H. Cohen and F. Strömberg, "Modular Forms: A Classical Approach", Graduate Studies in Mathematics, American Mathematical Society, 2017.
- J.S. Milne, "Modular Functions and Modular Forms", online course notes: http://www.jmilne.org/math/CourseNotes/mf.html
- T. Miyake, "Modular Forms", Springer-Verlag, 1989.
- Docent: Peter Bruin
- Docent: Eugenia Rosu