Workgroup on Number Theory and Algebraic Geometry

Secretariat
Kollegiengebäude Mathematik (20.30)
Room 1.027

Englerstraße 2, 76131 Karlsruhe

Office hours:
Mo - Fr, 9.15 - 11.45

Tel.: +49 721 608 43041

Fax.: +49 721 608 44244

# p-adic modular forms (Winter Semester 2013/14)

 Lecturer: PD Dr. Fabian Januszewski Lecture (0113100), Problem class (0113200) 4+2 Mathematics (from 5. semester)

 Lecture: Problem class: Wednesday 9:45-11:15 1C-03 Thursday 9:45-11:15 1C-03 Thursday 11:30-13:00 Z2
 Lecturer PD Dr. Fabian Januszewski Office hours: jederzeit wenn ich da bin. Room 1.023 Kollegiengebäude Mathematik (20.30) Email: januszewski@kit.edu Dr. Tobias Columbus Office hours: Friday, 10:30 - 12:00 Room 1.024 Kollegiengebäude Mathematik (20.30) Email: tobias.columbus@kit.edu

## Problem Sheets

See german page for problem sheets.

## Content

Modular forms and their -adic avatars play a central role in modern number theory. Despite being of analytic origin, modular forms contain valuable arithmetic information. A prominent example of this principle is Andrew Wiles' proof of Fermat's Last Theorem. The arithmeticity of modular forms is also reflected in the existence of -adic modular forms. The latter are more accessible to algebraic methods and are ubiquitious in modern number theory. Also, as they can be deformed -adically, in contrast to classical modular forms.

The close connection between modular forms and Galois representation becomes even closer in the -adic world.

In this class we will learn what a -adic modular form is, and how we can interpret classical modular forms as -adic ones. We will see that -adic modular forms can always be deformed, which has far reaching consequences. In particular it shows that congruences between classical modular forms occur abundantly.

In the lecture we will focus on the ordinary case, i.e. we will study Hida families. There are two approaches to this theory, one via moduli stacks of elliptic curves and geometric modular forms, and another via cohomology of arithmetic groups. We will follow the latter approach, as it is more easily accessble and requires less prerequisites.