Seminare und Vorträge im SS 2024

am Montag, 08. April :

Oberseminar Zahlentheorie

Johann Franke
Title: On the Proportion of Coprime Fractions in Number Fields
Abstract: We determine the asymptotic density of coprime fractions in those of the reduced fractions of number fields. When ordered by norms of denominators, we count a fraction as soon as it "appears" for the first time and no later. The natural density of coprime fractions in the set of reduced fractions may then be computed using well- known facts about Hecke L-functions.

14:00 Übungsraum 2 des Mathematischen Instituts

am Montag, 15. April :

Oberseminar Zahlentheorie

Jan-Willem van Ittersum (1)
Title: Lecture series on q-brackets
Abstract: In this lecture series, we study the generating series of several functions on partitions, that is q-brackets of functions on partitions. In the first lecture, we prove the Bloch-Okounkov theorem, stating that for the class of so-called shifted symmetric functions, such generating series are quasimodular. Then, in the second lecture, we how the study of this subject is being motivated by enumerative geometry. In particular, we study certain functions on partitions naturally associated to representations of symmetric groups, and a second family of functions on partitions related to hook-lengths of partitions. The last lecture will be devoted to results of the speaker, in which q-brackets are related to (generalized) MacMahon series and multiple zeta values.

14:00 Übungsraum 2 des Mathematischen Instituts

am Montag, 22. April :

Oberseminar Zahlentheorie

Jan-Willem van Ittersum (2)
Title: Lecture series on q-brackets
Abstract: See part (1).

14:00 Übungsraum 2 des Mathematischen Instituts

am Montag, 29. April :

Oberseminar Zahlentheorie

Anna Pippich
Title: Sup-norm bounds for Jacobi cusp forms
Abstract: In this talk, we report on new sup-norm bounds for the pointwise Petersson norm for Jacobi forms of integral weight \(k\) and integral index \(m\) for \(\mathrm{SL}_{2}(\mathbb{Z})\), which are normalized with respect to the Petersson inner product. For the proof, we essentially use the representation of the Jacobi cusp forms under consideration as linear combinations of modular forms of weight \(k-\frac{1}{2}\) for some congruence subgroup of \(\mathrm{SL}_{2}(\mathbb{Z})\) (depending on \(m\)) and suitable Jacobi theta functions; we then need to derive bounds for the pointwise Petersson norms of these functions to arrive at our result. This is joint work with A. Aryasomayajula and J. Kramer.

14:00 Übungsraum 2 des Mathematischen Instituts

am Montag, 06. Mai :

Oberseminar Zahlentheorie

Jan-Willem van Ittersum (3)
Title: Lecture series on q-brackets
Abstract: See part (1).

14:00 Übungsraum 2 des Mathematischen Instituts

am Montag, 13. Mai :

Oberseminar Zahlentheorie

Thomas Driscoll-Spittler
Title: Reflective modular varieties and their cusps
Abstract: Automorphic forms for the orthogonal group generalise classical modular forms. A reflective automorphic form has the special property that its divisor is supported on hyperplanes corresponding to roots. Often they are automorphic products and naturally appear in the theory of moduli spaces and in the representation theory of infinite- dimensional Lie algebras. It turns out that under certain regularity assumptions there are exactly 11 such reflective automorphic products of singular weight. The corresponding modular varieties have a very rich geometry. We show that their 1-dimensional type-0 cusps are naturally parametrised by the root systems obtained by Schellekens in his classification of holomorphic vertex operator algebras of central charge 24. This is joint work with Nils Scheithauer and Janik Wilhelm.

14:00 Übungsraum 2 des Mathematischen Instituts

am Montag, 27. Mai :

Oberseminar Zahlentheorie

Gilles Felber
Title: A restriction norm problem for Siegel modular forms
Abstract: Restriction norms are tools for measuring the equidistribution of functions such as eigenvalues of the Laplacian. In an arithmetic setting, this can often be expressed in terms of values of L-functions. We consider the case of the L²-norm of a Siegel modular form restricted to the imaginary axis. We establish an asymptotic formula on average with a power-saving error term. The Dirichlet series appearing in this problem has no Euler product and therefore does not belong to the Selberg class. Nevertheless, the result is shown to be consistent with the Mass Equidistribution Conjecture and the Lindelöf Hypothesis.

14:00 Übungsraum 2 des Mathematischen Instituts

am Montag, 03. Juni :

Oberseminar Zahlentheorie

Manuel Müller
Title: The basis problem for modular forms for the Weil representation
Abstract: The vector valued theta series of a positive-definite even lattice is a modular form for the Weil representation of \(\mathrm{SL}_2(\mathbb{Z})\). We show that the space of cusp forms for the Weil representation is generated by such functions. This gives a positive answer to Eichler's basis problem in this case. As applications we derive Waldspurger's result for scalar valued modular forms and give a new proof of the surjectivity of the Borcherds lift based on the analysis of local Picard groups.

14:00 Übungsraum 2 des Mathematischen Instituts

am Montag, 17. Juni :

Oberseminar Zahlentheorie

Catherine Cossaboom
Title: Patterns of primes in joint Sato-Tate distributions
Abstract: For \(j=1,2\), let \(f_j(z) = \sum_{n=1}^{\infty} a_{j}(n) e^{2\pi i nz}\) be a holomorphic, non-CM cuspidal newform of even weight \(k_j \ge 2\) with trivial nebentypus. For each prime \(p\), let \(\theta_{j}(p)\in[0,\pi]\) be the angle such that \(a_j(p) = 2p^{(k-1)/2} \cos \theta_{j}(p)\). The now-proven Sato--Tate conjecture states that the angles \( (\theta_j(p)) \) equidistribute with respect to the measure \(d\mu_{\mathrm ST} = \frac{2}{\pi}\sin^2\theta\,d\theta\). We show that, if \(f_1\) is not a character twist of \(f_2\), then for subintervals \(I_1,I_2 \subset [0,\pi]\), there exist infinitely many bounded gaps between the primes \(p\) such that \(\theta_1(p) \in I_1\) and \(\theta_2(p) \in I_2\). We also prove a common generalization of the bounded gaps with the Green--Tao theorem.

14:00 Übungsraum 2 des Mathematischen Instituts

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