Arbeitsgruppe Algebra und Zahlentheorie, MI Köln

Seminare und Vorträge im SS 2025

am Montag, 14. April :

Oberseminar Zahlentheorie

Pengcheng Zhang
Title: Elliptic curves and coefficients of meromorphic modular forms
Abstract: Coefficients of modular forms have been one of the main research topics in the theory of modular forms. In this talk, we will discuss our ongoing work, mostly numerical observations, on the properties of coefficients of meromorphic modular forms, which are closely related to elliptic curves. In particular, it seems that certain meromorphic modular forms behave numerically like symmetric powers of certain elliptic curves. If time permitted, we will also briefly discuss the proof of some of the numerical observations.

14:00

Übungsraum 2 des Mathematischen Instituts

am Montag, 28. April :

Oberseminar Zahlentheorie

Rashi Lunia
Title: Extreme values of L-functions
Abstract: In this talk, we will report on joint work with Gun, where we study extremal values of L-functions of Hecke eigenforms and Dirichlet-type L-functions. This builds upon earlier works of Soundararajan (2008) and Gun--Kohnen--Soundararajan (2023).

14:00

Übungsraum 2 des Mathematischen Instituts

am Montag, 19. Mai :

Oberseminar Zahlentheorie

Atul Dixit
Title: The Rogers-Ramanujan dissection of a theta function
Abstract: Page 27 of Ramanujan's Lost Notebook contains a beautiful identity which, as shown by Andrews, not only gives a famous modular relation between the Rogers-Ramanujan functions G(q) and H(q) as a corollary but also a relation between two fifth order mock theta functions and G(q) and H(q). In this talk, we will discuss a generalization of Ramanujan's relation which gives an infinite family of such identities. Our result shows that a theta function can always be ``dissected'' as a finite sum of products of generalized Rogers-Ramanujan functions.
Several new and well-known results are shown to be consequences of our theorem, for example, a generalization of the Jacobi triple product identity and Andrews' relation between two of his generalized third order mock theta functions. As will be shown, the identities resulting from our main theorem for s>2 transcend the modular world and hence look difficult to be written in the form of a modular relation. Using asymptotic analysis, we also offer compelling evidence that explains how Ramanujan may have arrived at his generalized modular relation. This is joint work with Gaurav Kumar.

14:00

Übungsraum 2 des Mathematischen Instituts

am Montag, 26. Mai :

Oberseminar Zahlentheorie

Charlotte Dombrowsky
Title: Central Values of L-functions of Twisted Modular Forms
Abstract: Computing the central values of the L-function of a twisted modular form is a hard problem in general. In this talk, we present a result by Males, Mono, Rolen, and Wagner, who provide a criterion for predicting whether the product of two such L- functions is zero. Their approach relies on locally harmonic mass forms. While their result is proven for square-free levels, we extend their work to modular forms with general composite levels. We will discuss the key ideas behind the proof and present some examples.

14:00

Übungsraum 2 des Mathematischen Instituts