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401-3112-23L

Number Theory II: Introduction to Modular Forms

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.
VVZ CR n/a

Last Updated: 2026-07-21 00:35:37

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.

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