[Paper Review] Toroidal compactifications of integral models of Shimura varieties of Hodge type

Keerthi Madapusi Pera|arXiv (Cornell University)|Nov 7, 2012

Algebraic Geometry and Number Theory4 citations

TL;DR

This paper constructs toroidal and minimal compactifications for integral models of Shimura varieties of Hodge type, reducing the problem to understanding the integral models themselves. It establishes the canonicity of these compactifications at hyperspecial level primes and provides a new proof of Y. Morita's conjecture on good reduction of abelian varieties with anisotropic Mumford-Tate groups modulo center, using a novel rationality property of Hodge cycles under p-adic uniformization.

ABSTRACT

We construct projective toroidal compactifications for integral models of Shimura varieties of Hodge type. We also construct integral models of the minimal (Satake-Baily-Borel) compactification. Our results essentially reduce the problem to understanding the integral models themselves. As such, they cover all previously known cases of PEL type, as well as all cases of Hodge type involving parahoric level structures. At primes where the level is hyperspecial, we show that our compactifications are canonical in a precise sense. We also provide a new proof of Y. Morita's conjecture on the everywhere good reduction of abelian varieties whose Mumford-Tate group is anisotropic modulo center. Along the way, we demonstrate an interesting rationality property of Hodge cycles on abelian varieties with respect to p-adic analytic uniformizations.

Motivation & Objective

• To construct toroidal and minimal (Satake-Baily-Borel) compactifications for integral models of Shimura varieties of Hodge type.
• To reduce the problem of compactifying these integral models to understanding the models themselves, thereby covering all previously known PEL-type cases.
• To establish the canonicity of the constructed compactifications at primes where the level structure is hyperspecial.
• To provide a new proof of Y. Morita's conjecture on the everywhere good reduction of abelian varieties whose Mumford-Tate group is anisotropic modulo its center.
• To establish a rationality property of Hodge cycles on abelian varieties with respect to p-adic analytic uniformizations.

Proposed method

• Utilizes a symplectic embedding of the Hodge-type Shimura datum $(G,X)$ into a Siegel Shimura datum $(\operatorname{GSp}(V), \operatorname{S}^\pm(V))$ to lift integral models from the Siegel moduli space.
• Constructs the integral model $\mathcal{S}_K$ as the normalization of the Siegel integral model $\mathcal{S}_{{K}^{\dagger}}$ in the generic fiber $\operatorname{Sh}_K(G,X)$.
• Applies logarithmic Dieudonné theory and p-adic analytic uniformization to analyze Hodge cycles and their rationality properties over p-adic fields.
• Employs rigid analytic geometry and p-adic exponential maps to prove integrality and boundedness of logarithms of functions in the local ring of the uniformized space.
• Uses the finiteness of the normalization and noetherianity of the base ring to show that certain analytic functions on the uniformized space lift to integral elements in the local ring.
• Applies the theory of finite maps between rigid analytic spaces to deduce finiteness of the integral closure and hence integrality of the relevant functions.

Experimental results

Research questions

• RQ1 Can toroidal and minimal compactifications be constructed for integral models of Shimura varieties of Hodge type?
• RQ2 Is the constructed compactification canonical at primes where the level structure is hyperspecial?
• RQ3 Does the rationality of Hodge cycles under p-adic uniformization imply integrality or boundedness in the local ring?
• RQ4 Can the construction of compactifications be reduced to understanding the integral models themselves?
• RQ5 Does the new method provide a proof of Y. Morita’s conjecture on good reduction of abelian varieties with anisotropic Mumford-Tate groups modulo center?

Key findings

• The paper constructs toroidal and minimal compactifications for integral models of Shimura varieties of Hodge type, reducing the problem to understanding the integral models themselves.
• At primes where the level is hyperspecial, the constructed compactifications are canonical in a precise sense, independent of the choice of symplectic embedding.
• A new proof is given of Y. Morita’s conjecture on the everywhere good reduction of abelian varieties whose Mumford-Tate group is anisotropic modulo its center.
• An interesting rationality property is established: if the logarithm of a function on the p-adic uniformization is analytic, then some power of the function lies in the maximal ideal of the local ring.
• The construction covers all previously known cases of PEL type, and extends the theory beyond the PEL setting, including non-PEL cases such as those arising from GSpin groups.
• The method relies on p-adic analytic techniques, including the convergence of the p-adic exponential and finiteness of normalization in rigid analytic geometry, to prove integrality of Hodge cycles.