# Modular forms

"There are five fundamental operations of arithmetic: addition, subtraction, multiplication, division, and modular forms."

Modular forms are holomorphic functions on the complex upper half place, which obey a simple transformation law with respect to certain Möbius transformations.
On the one hand, modular forms have numerous applications in number theory, e.g. they play a central role in the proof of Fermat's last theorem. On the other hand, it is easy to compute with modular forms, for example with a computer algebra systems like SAGE.

Lectures:
Tuesdays 4-6 p.m. in WSC-O-3.46
Wednesdays: 4-6 p.m. in WSC-S-U-3.01

Exercise session:
Thursdays: 12-2 p.m. in WSC-S-U-3.02
Exercise sheet 1
Exercise sheet 2
Exercise sheet 3
Exercise sheet 4
Exercise sheet 5
Exercise sheet 6
Exercise sheet 7
Exercise sheet 8
Exercise sheet 9
Exercise sheet 10
Exercise sheet 11

Prerequisites:
Complex analysis, basic algebra (groups, rings, fields, Galois theory)

Literature:
J.H. Bruinier, G. Harder, G. van der Geer, D. Zagier: The 1-2-3 of modular forms
D. Bump: Automorphic forms and representations
A. Deitmar: Automorphe Formen
F. Diamond, J. Shurman: A first course in modular forms
H. Hida: Elementary theory of L-functions and Eisenstein series
T. Miyake: Modular forms
J.-P. Serre: A Course in Arithmetic
G. Shimura: Introduction to the arithmetic theory of automorphic forms