# RTG Seminar Summer 2024

## RTG Seminar Summer term 2024

The Thursday morning seminar (10:15-11:45 in **WSC-N-U-3.05**) 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.

11.4.2024 | Julian Quast | On local Galois deformation rings |

18.4.2024 | Yitong Wang (Orsay) | Multivariable $(\phi,\Gamma)$-modules and local-global compatibility |

25.4.2024 | Sebastian Bartling | Moduli spaces of nilpotent displays |

2.5.2024 | Ludvig Modin | Graded unipotent quotients over a base scheme |

23.5.2024 | Laurent Berger (ENS Lyon) | Bounded functions on the character variety |

6.6.2024 | RTG applicants | Public talks |

13.6.2024 | Guillermo Gamarra Segovia | Extension of Eisenstein-Kronecker classes |

20.6.2024 | Lukas Bröring | On Quadratic Euler Characteristics of Symmetric Powers |

27.6.2024 | – | Symposium Düsseldorf/Essen/Wuppertal |

4.7.2024 | ALGANT Master Students | Thesis Rehearsal |

11.7.2024 | Federica Santi | Good compactified Jacobians |

18.7.2024 | – | Program discussion: Research seminar Winter 24/25 |

## Abstracts

### Ludvig Modin: Graded unipotent quotients over a base scheme

We present a new proof of the existence of projective geometric quotients for actions of a graded unipotent group acting on a projective scheme for actions that do not have unipotent stabilizers on the attracting locus. The proof works over a base scheme and without assuming the action is linear, generalizing from the original theorem which works for linear actions on complex projective varieties. If time permits we will explain a generalization of this result to Harder-Narasimhan type strata of algebraic stacks, and how one can relax the unipotent stabilizer assumption.

### Laurent Berger: Bounded functions on the character variety

The character variety $X$ is a rigid analytic curve defined by Schneider and Teitelbaum, in their work on $p$-adic Fourier theory. Here is a natural question about it: what is the ring of bounded functions on $X$? This question seems to be more difficult than it appears at first sight. I will discuss it, as well as some related problems and results.

### Guillermo Gamarra Segovia: Extension of Eisenstein-Kronecker classes

Eisenstein-Kronecker classes were constructed by Kings and Sprang as special classes of equivariant cohomology of an abelian scheme with coefficients in the completion of its Poincaré bundle. In this talk, I will describe how their method can be used to construct cohomology classes for families of abelian schemes, and how one can give an explicit description of them as a specific Eisenstein-Hilbert real analytic series.

### Lukas Bröring: On Quadratic Euler Characteristics of Symmetric Powers

We present our recent preprint with Anneloes Viergever on quadratic Euler characteristics of symmetric powers of curves and some adjacent computations. More specifically, we will compute the quadratic Euler characteristic of the $n$-th symmetric power of a smooth, projective curve over a field not of characteristic $2$, and we will compute the quadratic Euler characteristic of $\operatorname{Sym}^2X$ for $X$ a smooth, projective scheme over a field of characteristic not $2$.

We will relate these computations to a question of compatibility with power structures studied by Pajwani-Pál.

### Federica Santi: Good compactified Jacobians

Given a curve $X$ over an algebraically closed field $k$ and an integer $d$, we can consider the moduli space of line bundles on $X$ of total degree $d$. In many cases this moduli space fail to be compact. In this talk we introduce the notion of “Good compactified Jacobian” as an open connected substack of the stack of rank $1$ torsion-free sheaves on $X$ which admit a proper good moduli space. We will focus on some nodal curves, called necklaces, whose dual graph is an $n$-gon. We give a classification of all the good compactified Jacobians of a necklace $X$ and show that the moduli space is again a necklace.