401-3110-21L  Student Seminar in Number Theory: Modular Forms

SemesterFrühjahrssemester 2021
DozierendeM. Schwagenscheidt
Periodizitäteinmalige Veranstaltung
LehrspracheEnglisch
KommentarNumber of participants limited to 26.


KurzbeschreibungSeminar on the basic theory of classical elliptic modular forms
LernzielIn the seminar we will learn about the basic theory of classical elliptic modular forms. We start with the action of the modular group on the complex upper half-plane by Moebius transformations and describe its fundamental domain. As first examples of modular forms, we will investigate Eisenstein series, Ramanujan's Delta function, the Dedekind eta function, and the modular j-invariant. We will show that the space of modular forms of a fixed weight is finite dimensional, and determine its dimension. We will also study Hecke operators and the Petersson inner product on spaces of modular forms, and the L-functions associated with modular forms. Towards the end of the seminar we will discuss some more advanced topics, such as differential operators and quasimodular forms, the CM values of the j-function, and the periods of modular forms.
SkriptLink
LiteraturCohen, Strömberg: Modular Forms: A Classical Approach
Diamond, Shurman: A first course in modular forms
Koblitz: Introduction to elliptic curves and modular forms
Koecher, Krieg: Elliptische Funktionen und Modulformen
Lang: Introduction to modular forms
Miyake: Modular forms
Serre: A course in arithmetic
Zagier: The 1-2-3 of modular forms

Lecture notes on modular forms, available online: Link
Voraussetzungen / BesonderesWe will need the fundamental results from complex analysis, and some elementary group theory.

The website of the seminar can be found at
Link