# Project A3: Moduli spaces of G-shtukas and the Langlands Program

## Principal investigator

## Participating scientists

- Dr. Esmail Arasteh Rad
- Simon Huesken
- Felix Roetting
- Tim Schauch
- Prof. Dr. Peter Schneider
- Rajneesh Kumar Singh
- Anna Weiss

## Summary

In this project we investigate moduli spaces in the arithmetic of function fields which are analogs of Shimura varieties and generalizations of Drinfeld's moduli spaces. We concentrate on good and bad parahoric reduction of these moduli spaces and their interplay with Rapoport-Zink spaces. Related is the theory of rigid analytic period morphisms whose images we study. There are interesting connections with the p-adic local Langlands program, which we also plan to develop.Let us give a more detailed introduction. Much effort is currently devoted to the investigation of Shimura varieties and their reduction theory. Analogous objects in the arithmetic of function fields are the moduli spaces of Drinfeld modules, Drinfeld shtukas, abelian t-sheaves, and principal F-bundles. The latter carry the additional operation of a reductive group G and we also call them G-shtukas. In this project we study the moduli spaces of abelian t-sheaves and G-shtukas and their reductions, in particular their bad reduction of parahoric type. We expect that a thorough group theoretic understanding for G-shtukas yields insights that may also clarify the situation for general PEL Shimura varieties. Moreover as indicated by Ngô, the above moduli spaces might be relevant for proving Langlands functoriality over function fields.

The moduli of the above global objects have local counterparts which are called Rapoport-Zink spaces. They are on the number field side moduli spaces for Barsotti-Tate groups and were constructed by Rapoport and Zink as formal schemes. In the analog for function fields, Barsotti-Tate groups are replaced by so-called local G-shtukas. Local G-shtukas arise from (global) G-shtukas by localization and completion similarly as Barsotti-Tate groups arise from abelian varieties as "p-adic completion", that is as limit over their p-power torsion. Rapoport-Zink spaces for local G-shtukas were constructed in a paper of Hartl and Viehmann. The generic fibers of Rapoport-Zink spaces are rigid analytic spaces which are mapped under period morphisms to rigid analytic period domains. The period morphisms are not surjective in general as counterexamples of the principal investigator show. Similar phenomena arise for the period morphisms for Φ-modules of Pappas and Rapoport. In this project we also investigate the structure of Rapoport-Zink spaces for local G-shtukas with parahoric level structure and the image of the period morphisms for Barsotti-Tate groups, for local G-shtukas, and for Φ-modules.

We further clarify the relation of the latter with the p-adic local Langlands program pursued in project A2. Both the structure of the moduli spaces of parahoric G-shtukas and Rapoport-Zink spaces of local G-shtukas is strongly related to the combinatorics of the group G and its affine building, yielding a connection to project B4.