Lehr-Moodle der Universität Duisburg-Essen

Informationen und Kommunikationsweg für alle, die im Wintersemester 2025/26 die Modulabschlussprüfung in Modul 2 (Soziale Arbeit als Profession I) bei Helen Dambach ablegen.

The course will be an introduction to the theory of modular forms. Modular forms are very special holomorphic functions on the upper half plane and play an important role in various fields of mathematics. In this course, we will define modular forms on the upper half plane, study the modular curves as Riemann surfaces defined by congruence subgroups, explore the structure of the spaces of modular forms, define Hecke operators, Petersson products and study the theory of old forms and newforms. Then we will define L-functions for eigenforms and study the Rankin-Selberg method for a pair of modular forms. The course is designed for students with background in basic complex analysis and familiar with basic notions of algebra and aims to open the door for deeper topics in modern number theory and arithmetic geometry.

The course will introduce automorphic forms on GL_2(A_Q). We will first introduce adeles A_Q over the rational numbers Q and Fourier analysis on it. Then we will introduce the automorphic forms on GL_1(A_Q) (the ideles over Q), which is a generalisation of Dirichlet characters. After that we will move to GL_2(A_Q). We will generalise the notion of modular forms to automorphic functions and study Maass forms and their Fourier expansions. Hecke operators will then be defined in an adelic way. Then we will introduce automorphic forms on GL_2(A_Q) and automorphic representations of GL_2(A_Q).

Prerequisites: the course Modular Forms-1 (chapters 1-5 of Diamond-Shurman, A first course in modular forms; in particular the definitions of modular forms and Hecke operators).

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