Seminare und Vorträge im WS 2023/2024

am Montag, 23. Oktober :

Oberseminar Zahlentheorie

Ksenia Fedosova
Title: Divisor functions coming from string theory
Abstract: In this talk, we discuss inhomogeneous Laplace equations on the modular surface motivated by graviton scattering. The boundary conditions of solutions hint towards a class of convolution-type identities involving divisor functions, one of them being as follows: for any \(n \in \mathbb{N}\), the following holds: \[ \sum_{\stackrel{n_1, n_2 \in \mathbb{Z} \setminus \{0\}}{n_1+n_2=n}}\sigma_0(n_1)\sigma_0(n_2) \Big[ 2 + \tfrac{n_2-n_1}{n} \log | \tfrac{n_1}{n_2} | \Big] = \sigma_0(n) (2-\log \left(4 \pi^2 |n|\right) ) . \] We discuss how to solve the ODE and prove respective identities involving divisor functions. This is a joint work with Kim Klinger-Logan and Danylo Radchenko.

14:00 Seminarraum 3 des Mathematischen Instituts

am Montag, 30. Oktober :

Oberseminar Zahlentheorie

Prahlad Sharma
Title: Bilinear sums with \(GL(2)\) coefficients and the exponent of distribution of \(d_3\)
Abstract: We obtain the exponent of distribution \(1/2+1/30\) for the ternary divisor function \(d_3\) to square-free and prime power moduli, improving the previous results of Fouvry--Kowalski--Michel, Heath-Brown, and Friedlander--Iwaniec. The key input is certain estimates on bilinear sums with \(GL(2)\) coefficients obtained using the delta symbol approach.

14:00 Seminarraum 3 des Mathematischen Instituts

am Montag, 13. November :

Oberseminar Zahlentheorie

Michael Mertens
Title: Meromorphic Hecke Eigenforms
Abstract: We propose a notion of Hecke eigenforms for two families of quotient spaces of meromorphic cusp forms on \(SL_2(\mathbb{ℤ})\). We show that each quotient space in the first (resp. second family) is isomorphic as a Hecke module to the space \(S_{2k}\) (resp. \(M_{2k}\)) of cusp forms (resp. holomorphic modular forms) of the same weight on \(SL_2(\mathbb{ℤ})\). This is joint work with Kathrin Bringmann and Ben Kane.

14:00 Seminarraum 3 des Mathematischen Instituts

am Montag, 20. November :

Oberseminar Zahlentheorie

Krystian Gajdzica
Title: The log-behavior of the \(A\)-partition function \(p_A(n)\)
Abstract: The \(A\)-partition function \(p_A(n)\) represents the number of partitions of $n$ with parts in a given mulitset \(A\) of positive integers. In this talk, we mainly focus on the case when \(A\) is finite, and discuss the log-behavior of \(p_A(n)\). More precisely, we investigate the Bessenrodt-Ono type inequality, the log-concavity, the \(r\)-log-concavity, the higher order Turán inequalities and the Laguerre inequalities for the \(A\)-partition function. Finally, we also present some generalizations of the obtained results and state a few open problems.

14:00 Seminarraum 3 des Mathematischen Instituts

am Montag, 27. November :

Oberseminar Zahlentheorie

Berend Ringeling
Title: The Zeta Mahler Function
Abstract: In this talk we discuss the zeta Mahler function, which is closely related to the Mahler measure. We show that every such function can be meromorphically continued to the whole complex plane. We give many examples, and explain the computational aspect of this problem.

14:00 Seminarraum 3 des Mathematischen Instituts

am Montag, 04. Dezember :

Oberseminar Zahlentheorie

Annika Burmester
Title: An introduction to multiple q-zeta values
Abstract: We present the algebra \(Z_q\) of multiple q-zeta values and its strong connections to quasi-modular forms and generating series of partitions. Motivated by the theory of multiple zeta values, we explicitly describe a generating set of \(Z_q\) and its relations. The elements in this generating set are called balanced multiple q-zeta values. In particular, this leads to a conjectural weight-grading on \(Z_q\) extending the one of quasi-modular forms. In the end, an approach to a Hopf algebra structure on (a quotient of) \(Z_q\) is explained.

14:00 Seminarraum 3 des Mathematischen Instituts

am Montag, 11. Dezember :

Oberseminar Zahlentheorie

Robert Pollack
Title: Distribution of L-invariants of modular forms
Abstract: The distribution of invariants of modular forms has been studied in many contexts. The Sato-Tate conjecture makes a precise prediction on the distribution of normalized Hecke-eigenvalues for modular forms. Here one fixes a form and varies the eigenvalue. One could also fix the eigenvalue and vary the form and still this invariant has a beautifully predictable distribution.
In this talk, we will discuss p-adic variants of these questions and investigate the distribution of the p-adic size of Hecke-eigenvalues leading to Gouvea's conjecture. Further, we will study a more mysterious p-adic invariant of a modular form, namely the L-invariant. We will give an overview of this invariant and ultimately state a conjecture about its p-adic distribution. This work is joint with John Bergdall.

14:00 Seminarraum 3 des Mathematischen Instituts

am Montag, 18. Dezember :

Oberseminar Zahlentheorie

Gabriele Bogo
Title: Supersingular abelian surfaces and orthogonal polynomials.
Abstract: An abelian surface in characteristic p>0 is called supersingular if it is isogenous to the product of two elliptic curves with trivial p-torsion. I will show that the supersingular locus of certain families of abelian surfaces can be described in terms of zeros of orthogonal polynomials. The proof is based on the theory of modular embeddings and twisted modular forms, which will be explain in the talk. This is joint work with Yingkun Li.

14:00 Seminarraum 3 des Mathematischen Instituts

am Montag, 22. Januar :

Oberseminar Zahlentheorie

Pierre Charollois
Title: On Eisenstein’s Jugendtraum for Complex Cubic Fields
Abstract: In the early 2000’s Ruijsenaars and Felder-Varchenko have studied a generalisation of the Euler gamma function. This remarkable elliptic gamma function is a multivariable meromorphic q-series that comes from mathematical physics. It satisfies modular functional equations under the group SL3(Z) which make it an analogue of the Jacobi theta function. In this work, we unveil the place that this function and its avatars play in number theory. Our main thesis is that these functions play the role of meromorphic modular functions in extending the theory of complex multiplication to complex cubic fields. In other words we propose a conjectural solution to Hilbert’s 12th problem for complex cubic fields (following a line of research actually initiated G. Eisenstein). We give a lot of numerical evidences that support this conjecture, and relate it to the Stark conjecture by proving an analogue of the Kronecker limit formula in this cubic setting. Joint work with Nicolas Bergeron and Luis Garcia.

14:00 Seminarraum 3 des Mathematischen Instituts

Archiv

Seminare und Vorträge im SS 2023
Seminare und Vorträge im WS 2022/2023
Seminare und Vorträge im SS 2022
Seminare und Vorträge im WS 2021/2022
Seminare und Vorträge im SS 2021
Seminare und Vorträge im WS 2019/2020
Seminare und Vorträge im SS 2019
Seminare und Vorträge im WS 2018/2019
Seminare und Vorträge im SS 2018
Seminare und Vorträge im WS 2017/2018
Seminare und Vorträge im SS 2017
Seminare und Vorträge im WS 2016/2017
Seminare und Vorträge im SS 2016
Seminare und Vorträge im WS 2015/2016
Seminare und Vorträge im SS 2015
Seminare und Vorträge im WS 2014/2015
Seminare und Vorträge im SS 2014
Seminare und Vorträge im WS 2013/2014
Seminare und Vorträge im SS 2013
Seminare und Vorträge im WS 2012/2013
Seminare und Vorträge im SS 2012
Seminare und Vorträge im WS 2011/2012
Seminare und Vorträge im SS 2011
Seminare und Vorträge im WS 2010/2011
Seminare und Vorträge im SS 2010
Seminare und Vorträge im WS 2009/2010
Seminare und Vorträge im SS 2009
Seminare und Vorträge im WS 2008/2009
Seminare und Vorträge im SS 2008 und früher