ERC Starting Grant  2011

The project "Moduli spaces of local G-shtukas" provides a novel approach to the local Langlands programme via a comprehensive investigation of local G-shtukas and their moduli spaces and the exploitation of their relations to Shimura varieties. Local G-shtukas are generalisations to arbitrary reductive groups of the local analogue of Drinfeld shtukas. They also are the function field counterpart of p-divisible groups. Hence moduli spaces of local G-shtukas are of great interest, in particular for the geometric realisation of local Langlands correspondences. Compared to p-divisible groups local G-shtukas have several advantages. They can be defined and studied for any reductive group, enabling a systematic use of group theoretic methods and promising unified results. Furthermore, their local description by elements of loop groups makes them more accessible than the description of p-divisible groups by Cartier theory or displays. Comparison theorems to p-divisible groups then provide a novel way to insight into their moduli spaces. The research plan of this project is subdivided into three strands which mutually benefit from each other: Firstly we want to understand the representations realised in the cohomology of moduli spaces of local G-shtukas in connection with the geometric local Langlands programme. Secondly, we study the geometry of the moduli spaces and investigate several natural stratifications. Finally, we build the bridge to Shimura varieties. On the one hand we explore the source of new results obtained by transferring methods developed for one of the two sides (Shimura varieties resp. moduli spaces of local G-shtukas) to prove similar assertions for the other. On the other hand we establish closer ties by proving direct comparison theorems.