[Paper Review] On integral models of Shimura varieties

G. Pappas|arXiv (Cornell University)|Mar 29, 2020

Advanced Algebra and Geometry35 references4 citations

TL;DR

This paper introduces a canonical integral model for Shimura varieties at parahoric level structure by defining a 'canonical' model via the existence of a ${\mathcal{G}}$-display with parahoric structure. It proves that in Hodge type cases under tameness hypotheses, the integral models constructed by Pappas and Kisin are canonical and independent of choices, using a new theory of displays with ${\mathcal{G}}$-structure.

ABSTRACT

We show how to characterize integral models of Shimura varieties over places of the reflex field where the level subgroup is parahoric by formulating a definition of a "canonical" integral model. We then prove that in Hodge type cases and under a tameness hypothesis, the integral models constructed by the author and Kisin in previous work are canonical and, in particular, independent of choices. A main tool is a theory of displays with parahoric structure that we develop in this paper.

Motivation & Objective

• To define a canonical integral model for Shimura varieties at primes where the level subgroup is parahoric.
• To resolve the long-standing problem of characterizing integral models globally and proving their independence from construction choices.
• To extend the notion of 'canonical' models beyond the smooth case at hyperspecial level to the singular, parahoric case.
• To establish that the integral models of Pappas and Kisin for Hodge type Shimura varieties at tame parahoric primes are canonical and choice-independent.
• To develop a theory of displays with ${\mathcal{G}}$-structure as a foundational tool for this characterization.

Proposed method

• Define a canonical integral model via the existence of a ${\mathcal{G}}$-display, a filtered Frobenius module with ${\mathcal{G}}$-structure, where ${\mathcal{G}}$ is the parahoric group scheme associated to the level subgroup.
• Construct the universal $p$-divisible group over the integral model and use its Dieudonné crystal to define a ${\rm GL}$-display.
• Use Frobenius-invariant tensors in the Dieudonné module to lift the ${\mathcal{G}}$-torsor structure from the special fiber to the $p$-adic completion.
• Construct a ${\mathcal{G}}$-torsor over the formal completion of the Shimura variety using isomorphisms of tensors between the Dieudonné module and a fixed lattice.
• Use the local model diagram and ${\mathcal{G}}$-equivariance to define the structure of the ${\mathcal{G}}$-display over the integral model.
• Apply the dictionary between Dieudonné theory and ${\mathcal{G}}$-displays to lift the Frobenius structure and verify the display axioms.

Experimental results

Research questions

• RQ1 Can a canonical integral model be uniquely characterized for Shimura varieties at parahoric level, where smooth reduction fails?
• RQ2 Is the integral model constructed by Pappas and Kisin for Hodge type Shimura varieties with parahoric level structure independent of the choices made in its construction?
• RQ3 Can the notion of 'canonical' model be extended from the smooth case at hyperspecial level to the singular, parahoric case?
• RQ4 Does the existence of a ${\mathcal{G}}$-display with parahoric structure uniquely characterize the integral model of a Shimura variety?
• RQ5 Can the theory of displays be generalized to include ${\mathcal{G}}$-structures for reductive groups over $\mathbb{Z}_p$?

Key findings

• The integral models of Shimura varieties with parahoric level structure constructed by Pappas and Kisin are canonical in the sense of Definition 7.1.3, meaning they support a unique associated $({\mathcal{G}},{\rm M}^{{\rm loc}})$-display.
• These models are independent of choices in their construction, as the canonical structure is uniquely determined by the existence of a ${\mathcal{G}}$-display.
• The existence of a locally universal associated system on the formal completion of the model implies the existence of a $({\mathcal{G}},{\rm M}^{{\rm loc}})$-display.
• The Dieudonné crystal of the universal $p$-divisible group over the model gives rise to a ${\rm GL}$-display that lifts to a ${\mathcal{G}}$-display via tensor compatibility.
• The ${\mathcal{G}}$-torsor structure on the formal completion is constructed via isomorphisms of tensors between the Dieudonné module and a fixed lattice, ensuring compatibility with the local model.
• The Frobenius structure on the Dieudonné module induces a well-defined ${\mathcal{G}}$-equivariant isomorphism, completing the display structure and verifying the canonical property.