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

am Montag, 16. Juni :

Oberseminar Zahlentheorie

Olivia Beckwith
Title: A modular framework for generalized Hurwitz class numbers
Abstract: We explore the modular properties of generating functions for Hurwitz class numbers endowed with level structure. Our work is based on an inspection of the weight \(1/2\) Maass-Eisenstein series of level \(4N\) at its spectral point \(s=3/4\), extending the work of Duke, Imamoğlu and Tóth in the level \(4\) setting. We construct a higher level analogue of Zagier's Eisenstein series and a preimage under the \(\xi_{1/2}\)-operator. We deduce a linear relation between the mock modular generating functions of the level \(1\) and level \(N\) Hurwitz class numbers, giving rise to a holomorphic modular form of weight \(3/2\) and level \(4N\) for \(N > 1\) odd and square-free. Furthermore, we connect the aforementioned results to a regularized Siegel theta lift as well as a regularized Kudla-Millson theta lift for odd prime levels, which builds on earlier work by Bruinier, Funke and Imamoğlu. This is joint work with Andreas Mono.

14:00 Übungsraum 2 des Mathematischen Instituts

am Montag, 30. Juni :

Oberseminar Zahlentheorie

Daniele Dorigoni
Title: Modular Features of Superstring Scattering Amplitudes: Generalised Eisenstein Series and Theta Lifts
Abstract: We briefly review how superstring scattering amplitudes produce infinite families of real-analytic modular forms with respect to \(SL(2,\mathbb{Z})\). In particular, we focus our attention towards generalised Eisenstein series which are real-analytic modular invariant solutions to certain inhomogeneous Laplace-eigenvalues equations. We conjecture that string theory selects only rational linear combinations of generalised Eisenstein series which are expressible by lattice-sums and which can be written as theta-lifts of Maass-lifted versions of the modular local polynomials introduced by Bringmann and Kane. This talk is based on the preprint 2501.03996 with Michael B. Green and Congkao Wen.

14:00 Übungsraum 2 des Mathematischen Instituts

am Montag, 07. Juli :

Oberseminar Zahlentheorie

Pavel Guerzhoy

14:00 Übungsraum 2 des Mathematischen Instituts

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