Vorträge des SFB 1442

I will give an introduction to Lue Pan?s approach to the study of p-adic completed cohomology of Shimura varieties, focusing on the example of quaternionic Shimura curves. I will then discuss applications to locally analytic representations of the multiplicative group of the non-split quaternion algebra over Qp. This talk is based on joint work with Zhenghui Li and Benchao Su.

Abstract: The Madsen-Weiss theorem describes the homology of mapping class groups of surfaces in the stable range and similarly Galatius' theorem describes the homology of Aut(F_n) in the stable range. Both theorems can be proven by determining the homotopy type of an appropriate bordism category using a "scanning" argument. I will describe an alternative approach, based on joint work in progress with Shaul Barkan, that computes the homotopy types of these categories by studying symmetric monoidal functors out of them. This will work in the more general framework of modular infinity-operads. As other examples of this approach we will also obtain the stable homology of 3-dimensional handlebodies and connected sums of S^1 x S^2, confirming two conjectures of Hatcher.

I will give an introduction to Lue Pan?s approach to the study of p-adic completed cohomology of Shimura varieties, focusing on the example of quaternionic Shimura curves. I will then discuss applications to locally analytic representations of the multiplicative group of the non-split quaternion algebra over Qp. This talk is based on joint work with Zhenghui Li and Benchao Su.

Main objects in function field arithmetic are Anderson A-modules and their A-motives and A-comotives (dual A-motives). Of particular interest are those which are "abelian", since their A- motives are finitely generated, and those which are "coabelian", since their A-comotives are finitely generated. U. Hartl showed that for an Anderson A-module that is abelian and coabelian, there is a perfect pairing between its motive and its comotive, and hence an isomorphism between the dual of the motive, and the comotive. The pairing, however, is not given explicitly, and Hartl asks for such an explicit description. It also remained open whether abelian A-modules are the same as coabelian A-modules. After introducing the main objects, we will explain the main tasks. We will show that indeed abelian and coabelian are equivalent notions, and provide an explicit pairing between the A-motive and the A-comotive when the Anderson A-module is abelian/coabelian. Due to the nature of the A-motive and A-comotive, we will deal a lot with non-commutative polynomial rings and non-commutative Laurent series fields.