# Oberseminar Winter term 2023/24

Die Vorträge finden jeweils donnerstags um 16:45 Uhr im **Raum WSC-N-U-3.05** (im Mathematikgebäude ) statt. Directions from the train station.

Der Tee findet ab 16:15 statt.

Alle Interessenten sind herzlich eingeladen!

The seminar takes place on Thursday, starting at 4:45pm. The duration of each talk is about 60 minutes.

Everybody who’s interested is welcome to join.

26.10.2023 | Sally Gilles (IAS, Princeton) | On compactly supported p-adic proétale cohomology of analytic varieties (online) |

02.11.2023 | Michael Rapoport (Univ. Bonn) | Fundamental lemma and Arithmetic Fundamental lemma for the whole spherical Hecke algebra |

09.11.2023 | Otto Overkamp (Univ. Düsseldorf) | A proof of Chai’s conjecture |

16.11.2023 | – | Q&A session for the research seminar |

23.11.2023 | David Schwein (Bonn) | New supercuspidals at bad primes |

30.11.2023 | Matei Toma (IECL, Nancy) | Schur polynomials and the Hodge-Riemann bilinear relations |

07.12.2023 | Aprameyo Pal (HRI Prayagraj) | Multivariable (phi,Gamma)-modules and Iwasawa theory |

14.12.2023 | Wiesława Nizioł (Sorbonne Université, Paris / CNRS) | tba |

21.12.2023 | N. N. | tba |

11.01.2024 | Georg Tamme (Universität Mainz) | tba |

18.01.2024 | Stefano Vigni (Università di Genova) | tba |

25.01.2024 | Lukas Braun (Innsbruck) | tba |

01.02.2024 | Giuseppe Ancona (Université de Strasbourg) | tba |

## Abstracts

### Sally Giles: On compactly supported p-adic proétale cohomology of analytic varieties

I will define the p-adic proétale cohomology with compact support for rigid analytic varieties and present some properties that it satisfies. In particular, I will discuss comparison theorems between the (compactly supported versions of) proétale and de Rham cohomologies.This is based on a joint work with P. Achinger and W. Niziol.

Meeting ID: 532 517 3167

Password: the first prime >200.

### Otto Overkamp: A proof of Chai’s conjecture

The base change conductor is an invariant which measures the failure of a semiabelian variety to have semiabelian reduction. It was conjectured by Chai that this invariant is additive in certain exact sequences. I shall report on recent joint work with Takashi Suzuki which implies this conjecture. Time permitting, I shall also discuss counterexamples to a generalisation of Chai’s conjecture.

### Michael Rapoport: Fundamental lemma and Arithmetic Fundamental lemma for the whole spherical Hecke algebra

The FL and the AFL for the unit element in the spherical Hecke algebra of the unitary group are recent theorems (of Z. Yun, W. Zhang, R. Beuzart-Plessis, resp. of W. Zhang, A.Mihatsch/W. Zhang, Z. Zhang). Here FL is a statement in $p$-adic harmonic analysis, whereas AFL is a statement in arithmetic geometry. I will discuss the extension of these statements to an arbitrary element in the spherical Hecke algebra. This is joint work with C. Li and W. Zhang.

### David Schwein: New supercuspidals at bad primes

Supercuspidal representations are the elementary particles in the representation theory of reductive p-adic groups and play an important role in number theory as local factors of cuspidal automorphic representations. Constructing such representations explicitly, via (compact) induction, is a longstanding open problem. Although the problem has been solved for large p, a solution remains out of reach in general. I’ll discuss work in progress joint with Jessica Fintzen towards constructing some of these missing supercuspidals when p is (very!) small.

### Matei Toma: Schur polynomials and the Hodge-Riemann bilinear relations

The Hard Lefschetz Theorem and the Hodge-Riemann bilinear relations show how special the algebraic topology of complex projective manifolds is and the particular role played herein by ample divisor classes. In this talk we will present recent joint work with Julius Ross, in which we discuss Schur classes of ample vector bundles and show that they enjoy the Hard Lefschetz property and the Hodge-Riemann property in the same way as powers of ample divisor classes. Some applications to positivity in the intersection theory of algebraic cycles will be equally touched upon.

### Aprameyo Pal: Multivariable (phi, Gamma)-modules and Iwasawa theory

In the first half of the talk, I recall the motivation and construction for multivariable (Phi, Gamma)-modules. In the second half of the talk (joint work, partly in progress, with Gergely Zabradi), I show how to pass to Robba-style versions via overconvergence. The group cohomology can also be computed from the generalized Herr complex over the multivariate Robba ring. If time permits, I will indicate how the analytic Iwasawa cohomology (computed also from a generalized Herr-complex) will hopefully be useful for the (re)formulation of Bloch-Kato exponential maps in this setting.