# 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.