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# Introduction to Modular Forms

This course will run from October to December 2005.

### Aims and Objectives:

Modular forms are classical analytic objects which were the subject of
much interest early in the last century. For some time their interest
appeared to have diminished, until remarkable connections with a huge
range of other areas in pure Mathematics were discovered. The most
celebrated is, of course, the role they played in the proof of Fermat's
last theorem, through the conjecture of Shimura-Taniyama-Weil that
elliptic curves are modular, but they also occur in group theory amongst
other things. Therefore there is much current interest in the theory of
modular forms. The aim of this course is to cover the classical theory
of modular forms.
### Syllabus

This will include such material as:

- Definitions of the modular group, congruence
subgroups, modular forms
(defined for congruence subgroups as well as SL_2(Z), a difference
from Serre's book)
- examples, Eisenstein series, lattice functions
- Some number theoretic applications
- space of modular functions
- expansions at infinity, zeroes and poles using contour integrals
- Hecke operators
- Some Atkin-Lehner theory
- Peterson inner product
- Eigenforms
- L-functions and some properties
- Theta functions.

### Problem Sheets

Here.
### Some Reading Material (and useful stuff on the web)

- Serre, A Course in Arithmetic,
- T. Miyake, Modular forms,
- Knapp, Elliptic Curves,
- Bump, Automorphic Forms,
- Milne, Modular
Functions and Modular Forms,
- Stein, Algorithms
for computing with modular forms
- Verrill's Fundamental Domain
Drawer
- Verrill's
course notes from Spring 2004
- The Modular Forms
Database, also written by Stein
- q-series
package for Maple, written by Frank Garvan
- List
of eta-products, from the L-functions
website
- Frazer
Jarvis's course notes for modular forms
- Igor
Dolgachev's course notes for modular forms
- Notes from a course given
by Ken Ribet in 1996

Return to Lloyd Kilford's
homepage
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