Research Seminar Algebra, Number Theory and Discrete Mathematics

-

Some ideas about p-adic Hodge theory

We will give a naive introduction to some of the main objects that play a role in  p-adic Hodge-Tate decomposition such as rigid analytic varieties and étale cohomology groups. We will then give an idea of a way of investigating Hodge symmetry in the p-adic setting.

Julian Demeio (LUH)

Hilbert Property for Hilbert modular surfaces

Joint work with Damián Gvirtz. We prove the Hilbert Property for all Hilbert Modular surfaces of base level of K3 type. As an application, we obtain a positive answer to the Inverse Galois Problem in some new cases: namely for PSL_(F_{p^2}) for p lying in a union of several arithmetic sequences.

Revisiting points of bounded height on the split quintic del Pezzo surface

I will report on recent advances in our understanding of the Manin-Peyre conjecture for rational and integral points on split quintic del Pezzo surfaces over global fields, initiated by joint work with Ulrich Derenthal. I will also discuss more refined questions such as equidistribution and second-order main terms, based on ongoing joint work with U. Derenthal and F. Wilsch, and J. Glas and L. Faisant.

Algebraicity of Hecke L-values and p-adic interpolation

(joint with J. Sprang) Deligne’s conjecture for special values of arithmetic L- functions states roughly that the critical L-values divided by certain periods are algebraic numbers. In this lecture we will explain the conjecture and a new technique of proving integrality results for Eisenstein series via equivariant coherent cohomology of abelian varieties. This has led to a complete proof of Deligne’s conjecture for algebraic Hecke characters, which has been open for nearly 50 years. Our techniques allow also to construct easily a p-adic L-function for ordinary p, which encodes congruences between these L-values. In the non-ordinary case, we are able to generalize the p-adic L function of Katz, Boxall and Schneider-Teitelbaum for the first time to higher dimensional abelian varieties.

Do 4.12.25

Local-global principles for semi-integral points on Markoff orbifold pairs

In this talk we shall discuss the status of local-global principles for semi-integral points on orbifold pairs of Markoff type. If time permits, we will discuss a way to count Markoff orbifold pairs that satisfy the semi-integral Hasse principle while the corresponding Markoff surface lacks integral points. This talk is based on a joint work with Justin Uhlemann.

Dmytro Rudenko (LUH)

Perfectoid rings: from initial motivation to broader utilization.

We will introduce perfectoid rings relying on some historical precursors and Scholze's original motivation, the latter being tilting equivalence. We will give examples where perfectoid rings appear in nature, and how they can be used for uniformization of abelian varieties.

The circle method for quadrics over function fields

We study the number of rational points on a quadric over a function field. Using the circle method, we will derive an explicit formula without an error term.

On the K(π,1)-problem for hyperplane arrangements

Understanding when complements of complex hyperplane arrangements are K(π,1)-spaces is a central problem linking topology, geometry, and combinatorics in arrangement theory. Over the decades, many remarkable results were obtained but still this property remains only partially understood, and notable conjectures persist. I will survey the current state with particular attention to unresolved problems, and I will present new connections among existing criteria as well as further developments concerning necessary and sufficient conditions.