[Paper Review] Coherent sheaves on the stack of Langlands parameters

Xinwen Zhu|arXiv (Cornell University)|Aug 7, 2020

Advanced Algebra and GeometryMathematics65 references21 citations

TL;DR

This paper proposes a categorical framework for the arithmetic Langlands program by introducing coherent sheaves on stacks of Langlands parameters in local and global function field settings. It formulates conjectures linking these sheaves to local-global compatibility, Hecke actions, and cohomology of Shimura and Rapoport-Zink spaces, with key results including a derived S = T conjecture and a cohomological formula for generalized Rapoport-Zink spaces.

ABSTRACT

We formulate a few conjectures on some hypothetical coherent sheaves on the stacks of arithmetic local Langlands parameters, including their roles played in the local-global compatibility in the Langlands program. We survey some known results as evidences of these conjectures.

Motivation & Objective

• To formulate a categorical version of the local arithmetic Langlands correspondence using coherent sheaves on stacks of Langlands parameters.
• To provide a geometric framework for local-global compatibility in the arithmetic Langlands program.
• To connect coherent sheaves on these stacks to the cohomology of Shimura varieties and Rapoport-Zink spaces via Hecke actions.
• To generalize the S = T theorem to derived Hecke algebras in the context of geometric Langlands.
• To establish a conjectural cohomological formula for generalized Rapoport-Zink spaces using ind-coherent sheaves and local systems.

Proposed method

• Constructs moduli stacks of local and global Langlands parameters in the function field setting using derived algebraic geometry.
• Introduces the stack of arithmetic Langlands parameters as a moduli space for continuous homomorphisms from Galois groups to Langlands dual groups.
• Defines coherent sheaves on these stacks as central objects governing the Langlands correspondence.
• Uses derived Hecke algebras and ind-coherent sheaves to relate the sheaves to cohomology of Shimura and Rapoport-Zink spaces.
• Applies the functorial framework of [49] and [27] to relate the sheaves to the geometrization of the local Langlands correspondence.
• Proposes a conjectural equivalence between derived categories of coherent sheaves and Hecke eigensheaves, generalizing the S = T principle.

Experimental results

Research questions

• RQ1 How can coherent sheaves on stacks of Langlands parameters provide a categorical formulation of the local arithmetic Langlands correspondence?
• RQ2 What is the precise relationship between these sheaves and the cohomology of Shimura varieties in the global function field setting?
• RQ3 How do derived Hecke algebras act on the cohomology of Shtukas and Shimura varieties, and how does this relate to the proposed sheaves?
• RQ4 Can a conjectural cohomological formula for generalized Rapoport-Zink spaces be derived from the sheaf-theoretic framework?
• RQ5 What is the role of the tame stable center in realizing the Hecke action on cohomology via the proposed sheaves?

Key findings

• Conjecture 4.60 proposes a natural map from Hom spaces of sheaves on the local stack to Hom spaces of cohomology, compatible with compositions and extending the S = T principle to derived Hecke algebras.
• The composition of the Hecke action through the proposed sheaves is conjectured to coincide with the natural Hecke action on the cohomology of Shimura varieties.
• Conjecture 4.62 provides a HK × WE × Gb(F)-equivariant isomorphism between compactly supported cohomology of Rapoport-Zink spaces and Hom spaces of sheaves, with the right-hand side concentrated in non-negative degrees.
• The conjecture implies that the cohomology of basic Rapoport-Zink spaces vanishes below the middle degree, consistent with general expectations.
• At isolated smooth points of the local stack, the right-hand side of the cohomological formula is expected to concentrate in degree zero.
• The framework is consistent with known results in the hyperspecial level case, as confirmed by [77, 83], and extends them to the derived setting.