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# RTG Seminar

The Thursday morning seminar will be the “Research Training Group Seminar” where members of the RTG (PhD students, post-docs) present their results. Sometimes, we also have speakers from other places. Depending on the number of speakers and on the proposed topic, a speaker could use one or two sessions.

If you are interested in giving a talk in this seminar, please email me at ulrich.goertz@uni-due.de or simply talk to me at some point.

Currently, the seminar is conducted online. Email ulrich.goertz@uni-due.de for the access information.

 Date Speaker Title 5.11.2020 Nicolas Dupré A $p$-adic analytic quantum Beilinson-Bernstein theorem 12.11.2020 Fangzhou Jin A Gersten complex on real schemes 19.11.2020 Nils Ellerbrock Integrality of Stickelberger elements 3.12.2020 Matteo Costantini The Chern classes of strata of abelian differentials 10.12.2020 Robin Witthaus Towards a mod-$p$ metaplectic local Langlands correspondence for ${\rm GL}_2(\mathbf{Q}_p)$ 17.12.2020 Lennart Gehrmann The Gross-Kohnen-Zagier theorem via $p$-adic uniformisation 14.1.2021 Paulina Fust Continuous cohomology and Ext groups 4.2.2021 Ran Azouri Motivic Euler characteristic of nearby cycles and a quadratic conductor formula

## Abstracts

### Nicolas Dupré: A $p$-adic analytic quantum Beilinson-Bernstein theorem

Recent developments in the study of $p$-adic representations of $p$-adic groups, due to Ardakov-Wadsley among others, introduced $p$-adic analytic $\mathscr D$-modules as a tool in order to study representations geometrically. The techniques they employ revolve around p-adic analogues of the celebrated Beilinson-Bernstein localisation theorem, which roughly asserts that there is an equivalence between representations of a semisimple Lie algebra and modules over the sheaf of differential operators on the corresponding flag variety. Meanwhile, quantum group-like objects have recently been introduced in the $p$-adic setting.

In this talk, we will explain what quantum groups are and how to adapt ideas from noncommutative algebraic geometry in order to define quantum analogues of $p$-adic analytic flag varieties. We’ll then see how to obtain a Beilinson-Bernstein localisation theorem in this setup.

### Fangzhou Jin: A Gersten complex on real schemes

Given a dualizing complex on a scheme, we construct a Gersten-type complex on the associated real scheme, and show that this construction establishes a bridge between coherent duality and the Verdier-type duality on real schemes. This is a joint work with Heng Xie.

### Robin Witthaus: Towards a mod-$p$ metaplectic local Langlands correspondence for ${\rm GL}_2(\mathbf{Q}_p)$

The mod-$p$ representation theory of ${\rm GL}_2(\mathbf{Q}_p)$ is well understood and its relation to representations of the absolute Galois group of $\mathbf{Q}_p$ has been worked out by various people. We investigate the genuine smooth admissible irreducible mod-$p$ representations of a certain topological two-fold cover $\widetilde{{\rm GL}_2(\mathbf{Q}_p)}$ of ${\rm GL}_2(\mathbf{Q}_p)$, defined by Kubota, and relate them to Galois representations.

### Lennart Gehrmann: The Gross-Kohnen-Zagier theorem via p-adic uniformisation

The Gross-Kohnen-Zagier theorem says that certain generating series of CM points are modular forms of weight $3/2$ in the Jacobian of the modular curve $X_0(N)$.

We give a new proof of this result for quaternionic Shimura curves that admit $p$-adic uniformisation by the Drinfeld halfplane. More precisely, we give an expression for the generating series as the ordinary projection of the first derivative of a $p$-adic family of positive definite ternary theta series. This is joint work with Lea Beneish and Henri Darmon.