Arbeitsgruppe Algebra und Zahlentheorie, MI Köln

Seminare und Vorträge im SS 2019

am Dienstag, 09. April:

Oberseminar Zahlentheorie

Jennifer Kupka (Technische Universität Darmstadt),
Mock Modular Forms and Traces of Singular Moduli
Abstract: Mock theta functions were first introduced in Ramanujan's last letter to Hardy in 1920. This letter contained definitions of 17 different functions, given as q-hypergeometric series, and various statements of their properties but unfortunately no proofs. Proving some of these assertions has turned out to be rather difficult. However, results by Sander Zwegers offer the possibility to interpret mock theta functions as the holomorphic part of certain harmonic weak Maass forms of weight 1/2, so that the theory of modular forms can be used to examine the properties of mock theta functions. In this talk we will see an application of the Millson theta lift, which is used to derive finite algebraic formulas for the coefficients of mock theta functions of order 5 and 7 in terms of traces of CM-values of certain weakly holomorphic modular forms.

14:00

Übungsraum 2 des Mathematischen Instituts

am Dienstag, 16. April:

Oberseminar Zahlentheorie

Paul Kiefer (Technische Universität Darmstadt),
Orthogonal Eisenstein Series of Singular Weight
Abstract: In this talk we will introduce Eisenstein series of singular weight for O(2,n) and will construct them using Borcherds lift with vector-valued non-holomorphic Eisenstein series. We investigate their behavior at the critical points to obtain holomorphic modular forms of singular weight. Moreover we study their behavior at the boundary.

14:00

Übungsraum 2 des Mathematischen Instituts

am Dienstag, 23. April:

Oberseminar Zahlentheorie

Linda Frey (University of Copenhagen),
Denominators of Igusa Invariants of Genus 2 Curves with CM Jacobian
Abstract: In this talk I will give a short introduction into Igusa invariants of genus 2 hyperelliptic curves. As with the genus 1 curves (elliptic curves) there is a modular view of the world and an algebraic one. We will learn about important results on the denominators of the Igusa invariants and I will state conjecture which is a work in progress and a generalization of a result of Bilu - Habegger - Kühne. I will talk about several approaches to prove the conjecture and will be happy to talk about ideas of the audience after the talk.

14:00

Übungsraum 2 des Mathematischen Instituts

am Dienstag, 07. Mai:

Oberseminar Zahlentheorie

Jan-Willem van Ittersum (Utrecht University),
Quasimodularity of border strip moments
Abstract: Many functions on partitions give rise to quasimodular forms by taking a normalized generating series, called the q-bracket. A particular example are the hook-length moments, studied by Chen, Möller and Zagier. We introduce border-strip moments of which the hook-length moments are an example and show that their q-brackets are quasimodular forms. By doing so we encounter an interesting product on the set of all functions on partitions.

14:00

Übungsraum 2 des Mathematischen Instituts

am Dienstag, 14. Mai:

Oberseminar Zahlentheorie

No seminar (Conference in Romania)

am Dienstag, 21. Mai:

Oberseminar Zahlentheorie

Ulf Kühn (Universität Hamburg),
Multiple zeta values and q-series

14:00

Übungsraum 2 des Mathematischen Instituts

am Dienstag, 28. Mai:

Oberseminar Zahlentheorie

Nikolaos Diamantis (University of Nottingham),
Period functions of real analytic modular forms
Abstract: Real analytic modular forms are modular objects recently introduced by F. Brown and whose behaviour is a hybrid of the behaviour of weakly holomorphic forms and that of Maass cusp forms. We investigate some classical number theoretic aspects of those forms, including L-functions and period functions. As a "proof of concept" we show that, in the special case of weakly holomorphic forms, the "critical values" of our L-functions satisfy similar algebraicity relations as the holomorphic classical cusp forms. (Joint work with J. Drewitt).

14:00

Übungsraum 2 des Mathematischen Instituts

am Dienstag, 04. Juni:

Oberseminar Zahlentheorie

Cecilia Salgado (MPI Bonn),
Mordell Weil rank jumps in families of elliptic curves and the Hilbert property
Abstract: In this talk I will discuss the behaviour of the Mordell-Weil rank in families of elliptic curves. I'll start reviewing standard conjectures and previous results. In the second part, I will report on a work in progress with Dan Loughran in which we study rank jumps of the Mordell-Weil groups of the fibres of elliptic surfaces. I will present the cases for which we show that the set for which the rank jumps is not thin (in the sense of Serre).

14:00

Übungsraum 2 des Mathematischen Instituts

am Dienstag, 11. Juni:

Oberseminar Zahlentheorie

No seminar (Pentecoste)

am Dienstag, 18. Juni:

Oberseminar Zahlentheorie

Eric Hofmann (Universität Heidelberg),
A singular theta-lift for unitary groups and the construction of Green's forms
Abstract: In my talk, I want to present a singular theta-lift of Borcherds type for unitary groups U(p,q), and its application to the construction of Green's forms. This is joint work with Jens Funke. The singular theta-lift generalizes to unitary version of the Borcherds lift and is in a sense adjoint to the Kudla-Millson lift, quite like its analog in the orthogonal case studied by Bruinier and Funke. Using this lifting one can construct two families of Green's forms. Extending a recent result of Ehlen and Sankaran, we found that the generating series of the differences of the two Green's forms is a modular form. Further, we showed that the Green's form defined through the lift of Maass-Poincar\'{e} series satisfy an eigenvalue equations under the hyperbolic Laplacian and coincide with the objects by Oda and Tsusuki using completely different methods.

14:00

Übungsraum 2 des Mathematischen Instituts