## Advanced Topics in Algebra (V5A1) - Modular Forms and their associated Galois representations - Sommersemester 2015

Dr. Eugen Hellmann

E-mail: hellmann (add @math.uni-bonn.de)

Monday 10-12h, Wednesday 10-12h, MZ 0.006

First lecture: 08.04.2015

## Contents

Modular forms (respectively meromorphic modular forms) are holomorphic function on the complex upper half plane that have a prescribed transformation behavior under the action of the group SL_{2}(Z) and that admit a holomorphic (respectively meromorphic) extension to certain "cusps" of the upper half plane.

The space of modular forms admits a canonical action of so called Hecke-operators which form an algebra (the so called Hecke-algebra) and decompose the space of modular forms into a direct sum of eigenspaces for the Hecke action.

In the arithmetic theory one can reinterpret modular forms as sections of line bundles on the moduli space of elliptic curves (respectively on a compactification of this moduli space). The Hecke operators act (via correspondences) on these moduli spaces and hence a given system of Hecke eigenvalues (that is given by a cuspidal eigenform) will cut out a piece of the étale cohomology.

These Galois representations associated to cuspidal eigenforms (and higher dimensional analoga of this construction) play a fundamental role in modern number theory and arithmetic geometry.

Depending on the audience we hope to cover (a selection of) the following topics:

- Classical theory:
- Definition of meromorphic and holomorphic modular forms
- Eisenstein series and cusp forms
- Hecke-operators and the Hecke-algebra
- The Eichler-Shimura isomorphism
- Arithmetic theory:
- Introduction to moduli spaces of elliptic curves
- Modular forms as sections of line bundles on the moduli space of elliptic curves
- Hecke-correspondences
- Étale Cohomology of the moduli space of elliptic curves
- Construction of the Galois representations associated to a cuspidal eigenform

## Prerequisites

Complex Analysis and basic knowledge of singular (co-)homology, differential forms on (complex) manifolds; Algebraic geometry (roughly the amount of chapter II and III of Hartshorne's book). Some knowledge about elliptic curves is recommended.

## References

- P. Deligne:
*Formes modulaires et represéntations l-adiques*, Lecture Notes in Math. 179, Springer - P.Deligne, M. Rapoport
*Les schémas modules des courbes elliptiques*, in Modular Functions of One Variable II, Lecture Notes in Math. 349, Springer - N. Katz, B. Mazur:
*Arithmetic Moduli of elliptic curves*, Annals of Math. Studies 108, Princeton University Press - J.-P. Serre:
*A course in Arithmetic*, Graduate Texts in Math. 7, Springer

*Last modified: 01. 02. 2015, Eugen Hellmann*