Periodic Reporting for period 2 - NewtonStrat (Newton strata - geometry and representations)
Reporting period: 2019-12-01 to 2021-05-31
The most successful approach to prove instances of Langlands’ conjectures is via algebraic geometry, by studying suitable moduli (or parameter) spaces. Their cohomology carries actions both of a linear algebraic group (such as GLn) and a Galois group associated with the number field one is studying. A central tool in the study of the arithmetic properties of these moduli spaces is the Newton stratification, a natural decomposition based on the moduli description of the space. Recently the theory of Newton strata has seen two major new developments: Representation-theoretic methods and results have been successfully established to describe their geometry and cohomology. Furthermore, an adic version of the Newton stratification has been defined and is already of prime importance in new approaches within the Langlands programme.
This project aims at uniting these two novel developments to obtain new results in both contexts with direct applications to the Langlands programme, as well as a close relationship and dictionary between the classical and the adic stratifications. We investigate the geometry of Newton strata in loop groups and Shimura varieties, and representations in their cohomology. Secondly, we study corresponding geometric and cohomological properties of adic Newton strata. Finally, we establish closer ties between the two contexts.
For the second part of the project, we used new methods introduced by Fargues and Fontaine in p-adic Hodge theory to study the recently defined adic Newton strata. We determined which strata lie in the closure of a given one, and considered intersections with another interesting space, the so-called weakly admissible locus.