# 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