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Moduli spaces of local G-shtukas

Final Report Summary - G-SHTUKAS (Moduli spaces of local G-shtukas)

The Langlands programme is a large and prominent research field in pure mathematics. Langlands' conjectures link representations of absolute Galois groups of number fields or function fields of curves to the representation theory of a linear algebraic group G. The central method to realise Langlands correspondences is to use an investigation of the geometry and cohomology of suitable moduli spaces. In the arithmetic case these are Shimura varieties or moduli spaces of abelian varieties with additional structure, in the function field case one considers moduli spaces of Drinfeld shtukas with additional structure. Local questions on the latter moduli spaces are then studied via local G-shtukas and their moduli spaces. They are at the same time an analog of p-divisible groups. In this project we studied a broad range of questions concerned with moduli spaces of local G-shtukas:

- We extended the foundations of the theory, i.e. the construction of towers on the generic fiber of the moduli spaces, their cohomology groups and relations to moduli spaces of global G-shtukas.
- We studied the cohomology of the moduli spaces, both of the towers in the generic fiber and of the reduced schemes in the special fiber, so-called affine Deligne-Lusztig varieties. These questions are motivated by the goal to realize local Langlands correspondences.
- Our geometric results on the one hand give insight into fundamental properties of the special fibers of the moduli spaces such as non-emptiness, dimension, and the sets of connected components and of irreducible components. On the other hand we relate moduli spaces of local G-shtukas to corresponding moduli spaces of global G-shtukas via the Newton stratification. These results also led to a parallel theory for the geometry of Shimura varieties and moduli spaces of p-divisible groups.