[Paper Review] Rapoport-Zink Spaces For Local P-Shtukas and Their Local Models

Esmail Arasteh Rad|arXiv (Cornell University)|Jul 9, 2018

Algebraic Geometry and Number Theory27 references3 citations

TL;DR

This paper establishes local models for Rapoport-Zink spaces associated to local 𝔓-shtukas over function fields, completing a local analogue of the theory of local models for global 𝔈-shtukas. It proves the local model theorem, which provides a criterion for flatness and enables applications to Kottwitz-Rapoport stratifications, semi-simple trace of Frobenius, and cohomological cycle constructions on the generic fiber, endowing its cohomology with a module structure over finite-dimensional Chow rings.

ABSTRACT

In this article we first survey the analogy between Shimura varieties (resp. Rapoport-Zink spaces) and moduli stacks for global G-shtukas (resp. Rapooprt Zink spaces for local P-shtukas). This part is intended to enrich the dictionary between the arithmetic of number fields and function fields a bit further. Furthermore, to complete this picture, we also study some local properties of Rapoport-Zink spaces for local P-shtukas by constructing local models for them. This provides a "local" complementary to a previous work of the author, which was devoted to the study of the local models for the moduli stacks of global G-shtukas. We also discuss some of its applications.

Motivation & Objective

• To complete the analogy between Shimura varieties over number fields and moduli spaces of global 𝔈-shtukas over function fields by developing a local theory for local 𝔓-shtukas.
• To construct local models for Rapoport-Zink spaces of local 𝔓-shtukas, mirroring the local model theory for global 𝔈-shtukas established in prior work.
• To provide a local geometric framework that clarifies singularities and flatness properties of these Rapoport-Zink spaces over their reflex rings.
• To apply the local model theory to derive global consequences, such as stratifications and cohomological invariants, on the special and generic fibers.
• To endow the cohomology of the generic fiber of these spaces with a module structure over finite-dimensional Chow rings via cycle maps.

Proposed method

• Constructs local models for Rapoport-Zink spaces of local 𝔓-shtukas using a general setup where 𝔓 is a smooth affine group scheme over 𝔻 = 𝔽_q[[z]] with connected reductive generic fiber.
• Applies the local model theorem (Theorem 3.12) to establish a flat morphism from the local model to the reflex ring, ensuring flatness via modification of the naive model.
• Uses the local model diagram to define a correspondence between the special fiber of a boundedness condition and the generic fiber of the Rapoport-Zink space.
• Implements the Leray spectral sequence and decomposition theorem on Demazure resolutions of Schubert varieties to analyze cohomology of special fibers.
• Applies the motivic decomposition theorem and Leray-Hirsch theorem to show that the motive of a smooth Schubert variety is mixed Tate, implying finite-dimensional Chow groups.
• Composes the cycle class map with the pushforward of cycles to construct a module structure on the compactly supported ℓ-adic cohomology of the generic fiber over the Chow ring of the special fiber.

Experimental results

Research questions

• RQ1 How can local models for Rapoport-Zink spaces of local 𝔓-shtukas be constructed in the function field setting, analogous to the number field case?
• RQ2 What are the implications of the local model theorem for the flatness and singularities of these Rapoport-Zink spaces?
• RQ3 How can the local model theory be used to define stratifications on the special fiber, such as the Kottwitz-Rapoport stratification?
• RQ4 In what way does the local model theory facilitate the computation of the semi-simple trace of Frobenius on the cohomology of these spaces?
• RQ5 Can the cohomology of the generic fiber be endowed with a module structure over the Chow ring of the special fiber via cycle maps?

Key findings

• The local model theorem (Theorem 3.12) establishes a flat morphism from the true local model M_loc to the reflex ring, resolving the flatness issue present in the naive model M_naive.
• The construction of local models provides a criterion for flatness of Rapoport-Zink spaces for local 𝔓-shtukas over their reflex rings.
• The Kottwitz-Rapoport stratification on the special fiber of the Rapoport-Zink space is constructed via the local model diagram.
• The semi-simple trace of Frobenius on the cohomology of the generic fiber is related to that on Schubert varieties in twisted affine flag varieties, enabling cohomological computations.
• The compactly supported ℓ-adic cohomology of the generic fiber carries a module structure over the finite-dimensional Chow ring Ch*(Z) of the special fiber Z, when Z is irreducible and smooth.
• When the special fiber Z is smooth and irreducible, the motive M(Z) is a direct summand of a mixed Tate motive, implying that Ch^i(Z) is finite-dimensional for all i.