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401-3112-23L 7 Credits BSC , MSC D-MATH
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Number Theory II: Introduction to Modular Forms

Lecturers & Examiners: Prof. Dr. Emmanuel Kowalski
Be aware that there is a big overlap with former course units on modular forms, in particular with 401-4118-22L taught in the Spring Semester 2022 as an elective course. Only one of the two course units may be recognised for credits. More precisely, it is also not allowed to have recognised one course unit for the Bachelor's and the other course unit for the Master's degree.
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Last Updated: 2026-06-01 11:33:47

Abstract

This is an introduction to the theory of modular forms and itsapplications to number theory.

Objective

The goal of the course is to present in detail the basic theory and examples of classical modular forms together with a representative choice of applications to algebraic and analytic number number. More advanced applications and topics in direction of the Langlands program will also be sketched.

Content

The following topics will be presented: (1) Hyperbolic geometry and spaces of lattices. (2) Definition and examples of holomorphic modular forms, including Eisenstein series and theta series. (3) Spaces of modular forms. (4) Fourier coefficients of modular forms. (5) Hecke operators and their eigenvalues. (6) Hecke L-functions. (7) Applications: equidistribution of integral points on spheres, diophantine equations, estimates for class groups of quadratic fields.

Resources

Literature

J.P. Serre, "A Course in Arithmetic", Springer. H. Iwaniec, "Topics in Classical Automorphic Forms", AMS. N. Bergeron, "The spectrum of hyperbolic surfaces", Springer. T. Miyake, "Modular forms", Springer.

Learning Materials (Links)

General Information

Language
English
Levels
BSC , MSC
Frequency
Yearly recurring

Examination

Type
session examination
Mode
oral 30 minutes
The exam is offered only in the two examination sessions just after the course.

Course Components

Type Title Time & Place Hours
lecture with exercise Number Theory II: Introduction to Modular Forms
  • Wed 14:15-16:00 (HG E 1.1)
  • Fri 10:15-12:00 (HG E 1.1)
4 h weekly

Offered In