[Paper Review] Foliations and the cohomology of moduli spaces of bounded global $G$-shtukas

Stephan Neupert|arXiv (Cornell University)|Oct 19, 2016

Algebraic Geometry and Number Theory41 references9 citations

TL;DR

This paper establishes a decomposition of Newton strata in the special fiber of moduli spaces of bounded global G-shtukas into products of Rapoport-Zink spaces and Igusa varieties, both in the algebraic and adic settings. By constructing Igusa varieties via Id(bνi)-truncated isomorphisms and using uniformization morphisms, the authors relate the ℓ-adic cohomology of global G-shtukas to that of local and Igusa components, providing a framework for realizing Langlands correspondences through cohomological comparison theorems.

ABSTRACT

For arbitrary reductive groups $G$ defined over a finite field, we decompose Newton strata in the special fiber of moduli spaces of global $G$-shtukas into a product of Rapoport-Zink spaces and Igusa varieties. This allows us to compare the $\ell$-adic cohomology of these spaces together with the various actions on them.

Motivation & Objective

• To decompose Newton strata in the special fiber of moduli spaces of bounded global G-shtukas into products of Rapoport-Zink spaces and Igusa varieties.
• To construct Igusa varieties using Id(bνi)-truncated isomorphisms, avoiding reliance on the product decomposition of basic cases.
• To establish a comparison between the ℓ-adic cohomology of global G-shtukas and that of local and Igusa components via uniformization morphisms.
• To address cohomological challenges in adic analytic settings, particularly the failure of analytification to preserve strata and the behavior of vanishing cycles.

Proposed method

• Constructs Igusa varieties as moduli spaces parametrizing bounded global G-shtukas with Id(bνi)-truncated isomorphisms to the universal local G-shtuka (L+G, bνiσ*).
• Uses base change to the perfection of central leaves to enable direct Frobenius pullback and avoid complications from non-basic fundamental alcoves.
• Applies uniformization morphisms to relate the cohomology of global G-shtuka moduli spaces to products of Rapoport-Zink spaces and Igusa varieties.
• Employs analytification and vanishing cycle sheaves RΨan to compare cohomology in the adic setting, particularly for compactly supported cohomology.
• Analyzes the failure of analytification to preserve stratifications, showing that N(νi)an and N(νi)an,strat do not yield equivalent cohomology complexes in the Grothendieck group.
• Identifies obstructions in comparison theorems for Rapoport-Zink spaces, especially when formal schemes are not locally of finite type.

Experimental results

Research questions

• RQ1 Can Newton strata in the special fiber of moduli spaces of bounded global G-shtukas be decomposed into products of Rapoport-Zink spaces and Igusa varieties in the adic setting?
• RQ2 How can Igusa varieties be uniformly constructed for non-basic local G-shtukas without relying on product decompositions of basic cases?
• RQ3 To what extent do analytification and vanishing cycle sheaves preserve cohomological structures in the context of non-quasi-compact adic spaces?
• RQ4 Why do standard comparison theorems fail for Rapoport-Zink spaces that are not formal completions of schemes locally of finite type?
• RQ5 Can the cohomology of global G-shtukas be expressed as a virtual representation in terms of the cohomology of Rapoport-Zink and Igusa components?

Key findings

• The special fiber of the moduli space of bounded global G-shtukas decomposes as a product of Rapoport-Zink spaces and Igusa varieties, with the Igusa varieties represented as finite étale covers over the perfection of central leaves.
• Igusa varieties are constructed via Id(bνi)-truncated isomorphisms, providing a uniform moduli description that avoids the need for product decompositions in non-basic cases.
• The cohomology of the global G-shtuka moduli space is related to the cohomology of Rapoport-Zink and Igusa components through a uniformization morphism, yielding a comparison of virtual G(Aci) × ΓE′-representations.
• Analytification fails to preserve stratifications: the adic subspaces N(νi)an and the larger subsets N(νi)an,strat do not yield isomorphic cohomology complexes in the Grothendieck group.
• The sheaf of vanishing cycles RΨanη on Igusa varieties is not isomorphic to the constant sheaf Z/ℓrZ in the current setting, due to the failure of formal smoothness in the construction.
• Comparison theorems for Rapoport-Zink spaces fail in the current setting because the formal schemes are not locally of finite type, and the specialization morphism sp−1Uclε differs from the Berkovich space Uclrigε, invalidating key steps in prior proofs.