[Paper Review] Geometry of Shimura varieties of Hodge type over finite fields

Adrian Vasiu|ArXiv.org|Dec 11, 2007

Advanced Algebra and Geometry35 references3 citations

TL;DR

This paper provides a comprehensive, accessible overview of integral models of Shimura varieties of Hodge type over finite fields, focusing on their construction, moduli interpretations, smoothness, properness, and stratifications of special fibers. The key contribution is the establishment of the Traverso stratification and its purity property under certain conditions, with explicit bounds on the level $ n_v $ related to the $ p $-divisible group structure.

ABSTRACT

We present a general and comprehensive overview of recent developments in the theory of integral models of Shimura varieties of Hodge type. The paper covers the following topics: construction of integral models, their possible moduli interpretations, their uniqueness, their smoothness, their properness, and basic stratifications of their special fibres.

Motivation & Objective

• To provide a self-contained, accessible introduction to integral models of Shimura varieties of Hodge type for non-specialists.
• To clarify the construction, moduli interpretations, and geometric properties (smoothness, properness) of these integral models.
• To analyze the stratifications of the special fibers, particularly the Traverso stratification, and establish its purity property under key assumptions.
• To determine bounds on the level $ n_v $ for which the Traverso stratification stabilizes, especially in the supersingular case.

Proposed method

• Uses the theory of reductive groups, $ p $-divisible groups, $ F $-crystals, and Dieudonné modules to construct and analyze integral models.
• Applies deformation theory and Néron models to study moduli spaces of abelian schemes with Hodge cycles.
• Employs the language of $ ext{GL}(W) $-torsors and $ ext{Ker}( ext{H}(W(k)) \to \text{H}(W_m(k))) $ to define level $ m $ and Traverso stratifications.
• Relies on the canonical model $ \text{Sh}(G,\mathcal{X}) $ over a reflex field $ E(G,\mathcal{X}) $, and constructs integral models over $ O_{(v)} $ for primes $ v $.
• Uses the Newton polygon of $ D[p^{\lceil cd/(c+d)\rceil}] $ to determine the Newton polygon of $ (M,\phi) $, linking it to the stratification.
• Applies results from [Va5, Cor. 4.3] and [Va3, Main Thm. A] to prove regularity, equidimensionality, and quasi-affineness of the special fiber.

Experimental results

Research questions

• RQ1 Under what conditions does the level $ m $ stratification $ \mathfrak{L}_m $ of the special fiber of a Shimura variety of Hodge type have the purity property?
• RQ2 What is the smallest $ n_v $ such that the Traverso stratification $ \mathfrak{T} = \mathfrak{L}_{n_v} $ stabilizes, and how does it depend on the prime $ v $?
• RQ3 Can the Traverso stratification be explicitly described in terms of $ p $-divisible group truncations of level $ m $?
• RQ4 How do the isomorphism classes of $ \mathcal{E}_g[p^{n_v}] $ determine the isomorphism class of the $ p $-divisible group $ D $?
• RQ5 Is $ n_p = r $ in the case of supersingular points for $ \mathcal{A}_{r,1,N} $, as suggested by Traverso's conjecture?

Key findings

• The Newton polygon of a $ p $-divisible group $ D $ over a finite field is uniquely determined by $ D[p^{\lceil cd/(c+d)\rceil}] $, linking it to the stratification structure.
• The special fiber $ \mathfrak{n} $ of the integral model is regular, equidimensional, and quasi-affine over $ \mathcal{L}(N)_{v,l}^{\rm s} $, as shown via [Va5, Cor. 4.3].
• The Traverso stratification $ \mathfrak{T} = \mathfrak{L}_{n_v} $ exists and has the purity property under assumptions (*), (**), (***) and (***) on the group scheme.
• For supersingular points in $ \mathcal{A}_{r,1,N,\mathbb{F}_p} $, the Traverso stratification coincides with $ \mathfrak{L}_r $, implying $ n_p \geq r $.
• The identity $ \mathfrak{L}_m = \mathfrak{L}_{n_v} $ holds for all $ m \geq n_v $, showing stabilization of the stratification at level $ n_v $.
• The bound $ n_p \geq r $ is sharp in the supersingular case, and Traverso's conjecture suggests $ n_p = r $, though this remains unproven.