Special Fibres of Moduli Spaces of Shtukas

• Applicant: Patrick Bieker
• Subject Area: Mathematics
• Term: since 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 520675682

The study of the geometry of Shimura varieties at places of bad reduction is essential to the Langlands programme. For example, Harris-Taylor and Scholze used the cohomology of these spaces in order to establish the local Langlands correspondence for GL(n). The proposed project concerns the study of the geometry of certain moduli spaces of shtukas, the function field analogues of Shimura varieties, at places of bad reduction. Moduli spaces of shtukas were first introduced by Drinfeld for GL(n) and later generalised by Varshavsky for reductive groups and by Arasteh Rad and Hartl for arbitrary smooth affine group schemes. The goal of the project is to study how the irreducible components in the special fibre intersect in the case of Bruhat-Tits level structures. For parahoric level structures, the intersection behaviour is governed by the Kottwitz-Rapoport stratifcation (short: KR stratification). In this situation, there are (partial) results on the properties of the KR stratification, e.g., the KR strata are smooth and the closure relations among the strata are known. The novelty of the project is to go beyond the parahoric case using the applicant's construction of integral models for deeper Bruhat-Tits level structures. In a first step of the project, the special fibre of moduli spaces of shtukas is studied in more detail at parahoric level. For example, it is the goal to show the non-emptiness of the KR- (as well as the Newton-) strata also for shtukas in general. Moreover, an analogue of the Ekedahl-Kottwitz-Oort-Rapoport stratification - a refinement of the KR-stratification - is defined and studied. In a second step, a Kottwitz-Rapoport stratification is defined for deeper Bruhat-Tits level. It is the aim to give a characterisation of the natural index set and the closure relations for this stratification in terms of the combinatorics of the Bruhat-Tits building as in the parahoric case. This step relies on the one hand on the new results on the KR-stratification from the first step to deduce properties of the KR-stratification in general. On the other hand, the study of the general case is supported by explicit calculations in examples (in particular the Drinfeld case) using Drinfeld level structures for shtukas as introduced by the applicant. In future work, the results obtained in this project in the function field setting should then also be translated to the study of the geometry of Shimura varieties. Moreover, these results can then be used in the calculation of the trace of Frobenius on the nearby cycles on these models in future projects. This function is instrumental to the Langlands-Kottwitz method to determine the local factors of the L-function of the moduli space of shtukas (respectively the Shimura variety). The aim is to make this function explicit.

• DFG Programme: WBP Fellowship
• International Connection: USA
• Host: Professor Dr. Zhiwei Yun