| 12. April | Yingying Wang | Cohomology of the structure sheaf of Deligne-Lusztig varieties for GLn. |
| 19. April | Oliver Bräunling | Hilbert reciprocity using K-theory localization. The boundary map in K-theory localization at K_2 is the tame symbol. It sees the tamely ramified part of the Hilbert symbol, but no wild ramification. Gillet has shown how to prove Weil reciprocity using such boundary maps. This implies Hilbert reciprocity for curves over finite fields. However, phrasing Hilbert reciprocity for number fields in a similar way fails because it crucially hinges on wild ramification effects. I will explain how one can solve this problem (except at p=2) by introducing artificial singularities which "fatten up" K-theory with support in the special point. |
| 26. April | Yifei Zhao | What is a metaplectic group? The topological group SL_2(R) has a unique double cover up to isomorphism. It is not a linear algebraic group and yet plays a vital role in arithmetics, giving representation-theoretic meaning to modular forms of half-integral weights. This topological group is an example of a "metaplectic group". In this talk, I will propose a general algebraic framework for such groups over an arbitrary base scheme. They have a combinatorial classification and a notion of duality, extending those of reductive groups. Finally, we will catch a glimpse of Langlands duality for metaplectic groups. |
| 17. Mai | Hélène Esnault | Recent developments on rigid local systems. We shall review some of the general problems which are unsolved on rigid local systems and arithmetic $\ell$-adic local systems. We ‘ll report briefly on a proof (2018 with Michael Gröchenig) of Simpson's integrality conjecture for cohomologically rigid local systems. While all rigid local systems in dimension $1$ are cohomologically rigid (1996, Nick Katz), we did not know until last week of a single example in higher dimension which is rigid but not cohomologically rigid. We’ll present one series of examples (2022, Joint work with Johan de Jong and Michael Gröchenig). |
|
|
Marco D'Addezio | Boundedness of the p-primary torsion of the Brauer group of an abelian variety. I will present a new finiteness result for the p-primary torsion of the transcendental Brauer group of abelian varieties defined over finitely generated fields of positive characteristic p. This follows from a certain flat variant of the Tate conjecture for divisors which I formulated and proved for abelian varieties. At the end of the talk, I will also say some words about a second result, related to the main one, about the failure of the surjectivity of the specialisation morphism of the Néron–Severi group in families. More precisely, this theorem says that certain infinitely p-divisible p-torsion classes of the Brauer group of the abelian variety defined over the algebraic closure (which are not in the transcendental Brauer group by the main theorem) provide an obstruction to the surjectivity. |
| 14. Juni | Vytautas Paškūnas | Deformation Theory of local Galois representations. |
| 21. Juni | Christophe Spenke | On the sheaf cohomology of equivariant vector bundles on period domains over local fields |
| 28. Juni | Bogdan Zavyalov | Mod-p Poincare Duality in p-adic Analytic Geometry. Etale cohomology of F_p-local systems does not behave nicely on general smooth p-adic rigid-analytic spaces; e.g., the F_p-cohomology of the 1-dimensional closed unit ball is infinite. However, it turns out that the situation is much better if one considers proper rigid-analytic spaces. These spaces have finite F_p cohomology groups and these groups satisfy Poincare Duality if X is smooth and proper. I will explain how one can prove such results using the concept of almost coherent sheaves that allows to "localize" such questions in an appropriate sense and actually reduce to some local computations. If time permits, I will also mention some generalizations of these results in the relative setting (in progress). |
| 5. Juli | Damien Junger | De Rham cohomology of the tame cover in the Drinfel’d and Lubin-Tate towers Drinfeld has constructed two towers of covers $(\mathcal{M}^n_{LT})_n$ and $(\mathcal{M}^n_{Dr})_n$ as deformation spaces of formal modules. It is known that the supercuspidal part of the geometric étale $\ell$-adique cohomology with compact support of these spaces provides geometric realizations of local Langlands and Jacquet-Langlands correspondances. We would like to establish the same correspondances for De Rham cohomology with compact support. We will explain how we can proove these results in the particular case of the first cover by adapting the methods in Yoshida's and Wang's thesis where they exhibit a link between the geometry of these spaces and Deligne-Lusztig varieties. To relate the de Rham cohomology of these cover to the rigid cohomology of Deligne-Lusztig varieties, we will need to generalize an excision theorem of Grosse-Klönne. |
| 12. Juli | Mattias Strauch | TBA |