### Modular Forms (Winter Semester 2006/07)

- Lecturer: PD Dr. Stefan Kühnlein
- Classes: Lecture (1027)
- Weekly hours: 4
- Audience: mathematics, physics (5.-55. semester)

This lecture-series in english language will introduce to the theory of modular forms. It will mainly build upon complex analysis I, i.e. the residue theorem and its most basic consequences.

Schedule | |||
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Lecture: | Tuesday 9:45-11:15 | Seminarraum 12 | Begin: 24.10.2006, End: 16.2.2007 |

Friday 11:30-13:00 | Seminarraum 12 |

Building on complex analysis I we want to learn the most important basics in the theory of modular forms. These are holomorphic functions on the upper half-plane which satisfy a certain prescribed transformation law by under the action of Möbius-transformations from SL(2,Z) and a holomorphy-condition "at infinity".

The theory itself is an important vehicle in analytic number theory or arithmetic geometry. This will not really be significant for us, despite the fact that from time to time I will not be able to resist the wish make an excursion into one of these applications. I will not prove Wiles' Theorem!

Schedule of the lectures:

- elliptic functions and Eisensteinseries
- lattice classes and hyperbolic geometry
- modular forms for SL(2,Z)
- Hecke-Operators
- subgroups of SL(2,Z)
- miscellanea

The first two points will give a motivation for studying modular forms.

The lectures will be in english. Here is a list of results from complex analysis I will make use of during the course. By the way - I will also use basic group theoretic notions without much ado;-)

There is yet another announcement.pdf.

On tuesday, November 21, I made a **capital mistake** in my lecture which I will be going to correct on friday, November 24.

# Examination

There will be examinations at the end of Wintersemester. Exact dates and circumstances will have to be discussed once the lectures have started.

# References

- Apostol, Tom: Modular Functions and Dirichlet Series in Number Theory (Springer). This book stresses some elementary number theoretic applications of modular forms.
- Diamond, Fred; Shurman, Jerry: A First Course in Modular Forms (Springer). This book discusses modular forms with the aim to explain moularity of elliptic curves. In particular, its approach is very algebraic; this is very nice and assumes only basic facts from algebra, but I will stick more to classical stuff (and solve some of the exercises from this book). It is perfectly suitable for further reading and heartly recommended!
- Freitag, Eberhard; Busam, Rolf: Funktionentheorie I (Springer). This book contains many basics, in particular all copmplex analysis we will need is rigorously treated here.
- Knapp, Anthony: Elliptic Curves (PUP). Here the link between elliptic curves and modular forms is described.
- Koecher, Max; Krieg, Aloys: Elliptische Funktionen und Modulformen (Springer). The emphasis here is on elliptic functions.
- Lang, Serge: Introduction to Modular Forms (Springer). This is a classic, I grew up with this, and I hope you will be able to appreciate it, too.
- Lehner, Joseph: Discontinuous Groups and Automorphic Functions (AMS). This is even more classic, more geometric, and the focus on other discrete groups than the ones relevant to us is important in Riemannian geometry.
- Miyake, Toshitsune: Modular Forms (Springer). This is a very nice and readable introduction.
- Shimura, Goro: Introduction to the Arithmetic Theory of Automorphic Forms (PUP). Here comes the master himself. This book did pave the way towards a modern understanding of modular forms for a broader public, though it is written on a rather advanced level.