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

Semester | Spring Semester 2021 |

Lecturers | M. Schwagenscheidt |

Periodicity | non-recurring course |

Language of instruction | English |

Comment | Number of participants limited to 26. |

Abstract | Seminar on the basic theory of classical elliptic modular forms |

Objective | In 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. |

Lecture notes | Link |

Literature | Cohen, 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 |

Prerequisites / Notice | We will need the fundamental results from complex analysis, and some elementary group theory. The website of the seminar can be found at Link |