[Paper Review] Good and semi-stable reductions of Shimura varieties

Xuhua He, G. Pappas+1 more|arXiv (Cornell University)|Apr 25, 2018

Algebraic Geometry and Number Theory42 references3 citations

TL;DR

This paper establishes criteria for good and semi-stable reduction of Shimura varieties and their associated local models under tame ramification, proving that the local models constructed via the Pappas-Zhu framework are uniquely defined and have reduced special fibers. The key result identifies all cases of good or semi-stable reduction for local models of abelian-type Shimura varieties when the group splits over a tamely ramified extension, using the CCP condition and strict pseudo semi-stable reduction as central tools.

ABSTRACT

We study variants of the local models constructed by the second author and Zhu and consider corresponding integral models of Shimura varieties of abelian type. We determine all cases of good, resp. of semi-stable, reduction under tame ramification hypotheses.

Motivation & Objective

• To determine all cases of good and semi-stable reduction for Shimura varieties of abelian type with parahoric level structure at p.
• To establish the uniqueness of the local models constructed by Pappas and Zhu under tame ramification, resolving an open question about independence of auxiliary choices.
• To characterize the singularities of local models and relate them to the reduction type of the corresponding Shimura varieties.
• To introduce and analyze the 'CCP condition' as a key criterion for semi-stable reduction, and to prove its equivalence to rational strict pseudo semi-stable reduction.
• To provide a complete classification of local models with good or semi-stable reduction in the tame case, extending known results on modular curves to general Shimura varieties.

Proposed method

• Define local models as flat closures of the generic fiber of a naive model functor, using lattice chains and isotropic conditions in symmetric forms over p-adic fields.
• Use the framework of local model triples (G, {μ}, K) with G reductive over a p-adic field F, {μ} a minuscule cocharacter, and K a parahoric subgroup.
• Introduce the 'CCP condition' (compatibility of cocharacter and parahoric) as a necessary and sufficient condition for rational strict pseudo semi-stable reduction.
• Analyze the special fiber of the local model via explicit equations in affine charts, using coordinates on flag varieties and quadric constraints.
• Prove that the local model has reduced special fiber by showing that the defining equations (e.g., x1y2n + ... + xnyn + xn+1yn+1 = 0) hold universally via flatness.
• Use base change to unramified extensions to ensure well-definedness of the classification and to establish the equivalence between semi-stable reduction and the CCP condition.

Experimental results

Research questions

• RQ1 Under what conditions on the local model triple (G, {μ}, K) does the associated local model have good reduction?
• RQ2 When does the local model have semi-stable reduction, particularly under the assumption that G splits over a tamely ramified extension?
• RQ3 Is the local model construction independent of auxiliary choices, especially in the tame case?
• RQ4 What is the precise relationship between the CCP condition and the geometric structure of the special fiber?
• RQ5 Can the classification of good and semi-stable reduction be extended beyond the case of minimal or parahoric level structures?

Key findings

• The local model Mloc_K(G, {μ}) is uniquely defined and independent of auxiliary choices when G splits over a tamely ramified extension of F.
• The geometric special fiber of the local model is reduced and is isomorphic to the {μ}-admissible locus in the affine flag variety over the residue field k.
• The local model has good reduction if and only if the local model triple satisfies the CCP condition and the group is tamely ramified.
• For the case of nonsplit SO_{2n} with r = n and {0}, the local model is smooth, as shown by the Zariski closure of connected components in the affine chart being isomorphic to affine space.
• In contrast, for the split SO_{2n} case with μ = μ1, the local model does not have pseudo semi-stable reduction due to a singular intersection in the special fiber.
• The special fiber of the local model contains extra components when the naive model is not flat, as seen in the split SO_{2n} case with a P^{n-1} × P^{n-1} component not satisfying the defining equation.