The Bielefeld Algebraic and Arithmetic Geometry Seminar WS21/22

Welcome to the website of the Bielefeld Algebraic and Arithmetic Geometry Seminar! For more information, please contact one of the organizers: Claudia Alfes-Neumann, Eike Lau, Michael Spiess, Jeanine Van Order, Charles Vial, Thomas Zink, and also (from the Paderborn branch of the team) Fabian Januszewski.

Talks

Wednesday 1 June, 2022
14:15	V2 210/216	Yingkun Li (TU Darmstadt) Algebraicity of higher Green functions at CM points [Abstract] Algebraicity of higher Green functions at CM points. Abstract: By the classical theory of complex multiplication, the modular j-function takes algebraic values at CM points. It is an interesting question to ask about the algebraic nature of other types of automorphic functions at CM points. For the automorphic Green function at integral parameters, Gross and Zagier conjectured in the 1980s that their values at CM points are essentially logarithms of algebraic numbers. In this talk, we will discuss recent progress toward this conjecture and its generalization to the setting of orthogonal Shimura varieties. This is partly joint with Jan Bruinier and Tonghai Yang.
Wednesday 1 June, 2022
16:00	V2 210/216	Kuan-Wen Lai (Bonn University) Frobenius nonclassical hypersurfaces [Abstract] Frobenius nonclassical hypersurfaces are interesting examples of non-reflexive projective varieties in positive characteristics. They are well-studied in the case of curves, but not much is known in higher dimensions. In this talk, I will introduce sharp upper and lower bounds on the degrees of such hypersurfaces over finite fields as well as characterizations when their degrees attain maximal. This is a work in progress joint with Lian Duan and Shamil Asgarli.
Wednesday 12 January, 2022
10:30	ZiF	Vytautas Paskunas (Uni Duisburg-Essen) On local Galois deformation rings. [Abstract] We show that framed deformation rings of mod p representations of the absolute Galois group of a p-adic local field are complete intersections of expected dimension. We determine their irreducible components and show that they and their special fibres are normal and complete intersection. As an application we prove density results of loci with prescribed p-adic Hodge theoretic properties. (joint work with Gebhard Boeckle and Ashwin Iyengar. The preprint is available at https://arxiv.org/abs/2110.01638).
Wednesday 12 January, 2022
11:45	ZiF	Fabian Januszewski (Uni Paderborn) TBA [Abstract] TBA
Wednesday 12 January, 2022
14:00	ZiF	Ignacio Barros Reyes (Orsay) On the irrationality of moduli spaces of K3 surfaces [Abstract] I will talk about the problem of determining the birational complexity of moduli spaces of curves and K3 surfaces. I will recall some recently introduced invariants that measure irrationality and talk about what is known for these moduli spaces. In the second half, I will report on joint work with D. Agostini and K.-W. Lai, where we study how the degrees of irrationality of the moduli spaces of polarized K3 surfaces grow with respect to the genus g. We provide polynomial bounds. The proof relies on Kudla's modularity conjecture for Shimura varieties of orthogonal type. For special genera, we exploit the deep Hodge theoretic relation between K3 surfaces and special hyperkähler fourfolds to obtain much sharper bounds.
Wednesday 12 January, 2022
15:30	ZiF	Kestutis Cesnavicius (Orsay) Grothendieck--Serre in the quasi-split unramified case [Abstract] The Grothendieck--Serre conjecture predicts that every generically trivial torsor under a reductive group scheme G over a regular local ring R is trivial. We will explain how to settle it in the case when G is quasi-split and R is unramified.

Wednesday 24 November, 2021
14:30	H6	Thomas Geisser (Rikkyo University) Class field theory of schemes over local fields [Abstract] I will discuss recent work with B. Morin on a (conjectural) construction of a cohomology theory for varieties over local fields consisting of topological groups which satisfy a Poincare duality. I will present some unconditional results, and discuss application to abelian class field theory for varieties over local fields.
Wednesday 24 November, 2021
17:00	H6	Katharina Hübner (Heidelberg) Logarithmic differentials on discretely ringed adic spaces [Abstract] The object of interest in this talk is a certain subsheaf $\Omega^+_X$ of the sheaf of differentials $\Omega_X$ of a discretely ringed adic space $X$ over a field $k$. It is expected to play an important role in the tame cohomology theory of~$X$. The first part will be dedicated to an introduction to discretely ringed adic spaces. We will then define $\Omega^+_X$ using K\"ahler seminorms and establish a relation with logarithmic differentials. Finally we study the case where $X = Spa(U,Y)$ for a scheme $Y$ over $k$ and an open subscheme $U$ such that the corresponding log structure on $Y$ is log smooth. It turns out that $\Omega^+_X(X)$ equals $\Omega^{log}_{(U,Y)}(U,Y)$.

Wednesday

24 November, 2021

14:30

Thomas Geisser (Rikkyo University)
Class field theory of schemes over local fields

[Abstract]
I will discuss recent work with B. Morin on a (conjectural) construction of a cohomology theory for varieties over local fields consisting of topological groups which satisfy a Poincare duality. I will present some unconditional results, and discuss application to abelian class field theory for varieties over local fields.

Wednesday

24 November, 2021

17:00

Katharina Hübner (Heidelberg)
Logarithmic differentials on discretely ringed adic spaces

[Abstract]
The object of interest in this talk is a certain subsheaf $\Omega^+_X$ of the sheaf of differentials $\Omega_X$ of a discretely ringed adic space $X$ over a field $k$. It is expected to play an important role in the tame cohomology theory of~$X$. The first part will be dedicated to an introduction to discretely ringed adic spaces. We will then define $\Omega^+_X$ using K\"ahler seminorms and establish a relation with logarithmic differentials. Finally we study the case where $X = Spa(U,Y)$ for a scheme $Y$ over $k$ and an open subscheme $U$ such that the corresponding log structure on $Y$ is log smooth. It turns out that $\Omega^+_X(X)$ equals $\Omega^{log}_{(U,Y)}(U,Y)$.

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The Bielefeld Algebraic and Arithmetic Geometry Seminar WS21/22

Talks

Wednesday

1 June, 2022

Wednesday

1 June, 2022

Wednesday

12 January, 2022

Wednesday

12 January, 2022

Wednesday

12 January, 2022

Wednesday

12 January, 2022

Wednesday

24 November, 2021

Wednesday

24 November, 2021