The dimension of Oort strata of Shimura varieties of PEL-type.

*(English)*Zbl 1052.14026
Faber, Carel (ed.) et al., Moduli of abelian varieties. Proceedings of the 3rd Texel conference, Texel Island, Netherlands, April 1999. Basel: Birkhäuser (ISBN 3-7643-6517-X/hbk). Prog. Math. 195, 441-471 (2001).

From the introduction: The moduli space of principally polarized abelian varieties of dimension \(g\) admits in positive characteristic \(p\) a stratification by the \(p\)-rank of the abelian variety, i.e. by the rank of the \(p\)-torsion. In the case of the moduli space of elliptic curves this is simply the stratification given by the (open and dense) ordinary locus and the zero-dimensional supersingular locus. But for \(g>2\) this stratification is too coarse for many purposes. A refinement is provided by stratifying the moduli space according to the Newton polygon of an abelian variety. This stratification has been studied thoroughly by F. Oort [in: Moduli of abelian varieties. Prog. Math. 195, 417–440 (2001; Zbl 0982.68129)]. He also suggested to study another stratification given by the isomorphism type of the \(p\)-torsion which we will call the Oort stratification in the sequel (often it is also called Ekedahl-Oort stratification). This stratification is also a refinement of the \(p\)-rank stratification.

Both stratifications can also be considered for models of Shimura varieties of PEL-type which classify abelian varieties with certain additional structures. The precise definition of the Newton polygon stratification in this case together with a description of the strata which can occur and the specialization property of the stratification is due to M. Rapoport and M. Richartz [Compos. Math. 103, No. 2, 153–181 (1996; Zbl 0874.14008)]. It follows from results of A. J. de Jong and F. Oort [J. Am. Math. Soc. 13, No. 1, 209–241 (2000; Zbl 0954.14007)] that the codimension of a stratum in the closure of any non-empty stratum lying directly above can be only one. Combining these results one gets lower bounds for the dimension of the strata, but in general it is not known whether they are empty or not except for some special cases: In the case of the moduli space of principally polarized abelian varieties Oort shows that all strata are nonempty.

The Oort stratification has been examined in the case of Hilbert-Blumenthal varieties by E. Z. Goren and F. Oort [J. Algebr. Geom. 9, No. 1, 111–154 (2000; Zbl 0973.14010)] where they give a precise description of the occurring isomorphism classes and the dimension of the strata. Further they prove that the closure of a stratum is a union of strata and that all strata are quasi-affine. In the case of the moduli space of principally polarized abelian varieties Oort has obtained analogous results. Further, van der Geer computed the Chow classes of the Oort strata. For arbitrary good reductions of Shimura varieties of PEL-type, B. Moonen has given a description of the possible occurring strata in terms of the Shimura datum.

The main goal of this article is the definition and the calculation of the dimension of the Oort strata for good reductions of arbitrary Shimura varieties of PEL-type. We show that the closure of a stratum is the union of strata. To formulate the main result, let \(\underline X=(X,\lambda,\iota)\) be the \(p\)-torsion of an abelian variety together with a polarization and an action of a \(\mathbb{Z}_{(p)}\)-algebra unramified at \(p\), and denote by \(\xi\) its isomorphism class. If we denote by \(A_\xi\) the corresponding Oort stratum, we show

Theorem: For \(p>2\), \(A_\xi\) is locally closed and its codimension is equal to \(\dim(\underline {\operatorname{Aut}}(\underline X))\).

To do this, we follow an idea of A. J. de Jong which can be formulated roughly as follows: Define the moduli space B of truncated Barsotti-Tate groups with PEL-structure and show that the canonical morphism from the moduli space \(A\) of abelian varieties with PEL-structure into this moduli space is smooth. Define a smooth cover \(\widetilde B\) of \(B\) as the moduli space of truncated Barsotti-Tate groups with PEL-structure and with a trivialisation of its structure sheaf as a module over the base and denote by \(\widetilde A\) the fibre product of \(A\) with \(\widetilde B\) over \(B\). Then the inverse image of \(A_\xi\) in \(\widetilde A\) is the inverse image of a locally closed subscheme of \(\widetilde B\) of codimension \(\dim (\underline{\operatorname{Aut}} (\underline X))\) which implies the claim.

In order to calculate the dimension of \(\underline{\operatorname{Aut}} (\underline X)\) it would be preferable to calculate instead the dimension of the group of automorphisms of the Dieudonné space of \(\underline X\). For this we define Dieudonné spaces with PEL-structure over arbitrary schemes which allows us to speak of the moduli space of these Dieudonné spaces \(D\). Using the crystalline Dieudonné functor we get a morphism \(D\): \(B\to D\) which induces a homeomorphism of the underlying Zariski spaces. It follows that the stratification by isomorphism classes of Dieudonné spaces is equal to the Oort stratification. Further we get \(\dim(\underline {\operatorname{Aut}}(\underline X))= \dim (\underline{\text{Aut}} \bigl(D(\underline X)\bigr)\).

We now give an overview of the structure of this work: In the first chapter truncated Barsotti-Tate groups with PEL-structure and their moduli space are defined. In the second chapter we show that the canonical morphism of the deformation functor of an abelian variety (or its Barsotti-Tate group) into the deformation functor of its \(p^n\)-torsion is formally smooth for \(p>2\).

In the third chapter we deduce that the moduli space of truncated Barsotti-Tate groups is smooth over \(\mathbb{Z}_{(p)}\) and that its fibres are of dimension zero. In the fourth chapter we endow the set of isomorphism classes of truncated Barsotti-Tate groups with a topology using the general language of algebraic stacks and in the fifth chapter we relate truncated Barsotti-Tate groups and Dieudonné spaces and define a morphism between their respective moduli spaces and show that this is a homeomorphism. The sixth chapter finally gives the definition of the Oort stratification of a scheme \(S\) with respect to a truncated Barsotti-Tate group with additional structures over \(S\). In particular we define the Oort stratification of the model of a Shimura variety of PEL-type at a prime of good reduction and give the main theorem. After that we give some conditions when the Oort strata are nonempty. In the last chapter we give two examples.

For the entire collection see [Zbl 0958.00023].

Both stratifications can also be considered for models of Shimura varieties of PEL-type which classify abelian varieties with certain additional structures. The precise definition of the Newton polygon stratification in this case together with a description of the strata which can occur and the specialization property of the stratification is due to M. Rapoport and M. Richartz [Compos. Math. 103, No. 2, 153–181 (1996; Zbl 0874.14008)]. It follows from results of A. J. de Jong and F. Oort [J. Am. Math. Soc. 13, No. 1, 209–241 (2000; Zbl 0954.14007)] that the codimension of a stratum in the closure of any non-empty stratum lying directly above can be only one. Combining these results one gets lower bounds for the dimension of the strata, but in general it is not known whether they are empty or not except for some special cases: In the case of the moduli space of principally polarized abelian varieties Oort shows that all strata are nonempty.

The Oort stratification has been examined in the case of Hilbert-Blumenthal varieties by E. Z. Goren and F. Oort [J. Algebr. Geom. 9, No. 1, 111–154 (2000; Zbl 0973.14010)] where they give a precise description of the occurring isomorphism classes and the dimension of the strata. Further they prove that the closure of a stratum is a union of strata and that all strata are quasi-affine. In the case of the moduli space of principally polarized abelian varieties Oort has obtained analogous results. Further, van der Geer computed the Chow classes of the Oort strata. For arbitrary good reductions of Shimura varieties of PEL-type, B. Moonen has given a description of the possible occurring strata in terms of the Shimura datum.

The main goal of this article is the definition and the calculation of the dimension of the Oort strata for good reductions of arbitrary Shimura varieties of PEL-type. We show that the closure of a stratum is the union of strata. To formulate the main result, let \(\underline X=(X,\lambda,\iota)\) be the \(p\)-torsion of an abelian variety together with a polarization and an action of a \(\mathbb{Z}_{(p)}\)-algebra unramified at \(p\), and denote by \(\xi\) its isomorphism class. If we denote by \(A_\xi\) the corresponding Oort stratum, we show

Theorem: For \(p>2\), \(A_\xi\) is locally closed and its codimension is equal to \(\dim(\underline {\operatorname{Aut}}(\underline X))\).

To do this, we follow an idea of A. J. de Jong which can be formulated roughly as follows: Define the moduli space B of truncated Barsotti-Tate groups with PEL-structure and show that the canonical morphism from the moduli space \(A\) of abelian varieties with PEL-structure into this moduli space is smooth. Define a smooth cover \(\widetilde B\) of \(B\) as the moduli space of truncated Barsotti-Tate groups with PEL-structure and with a trivialisation of its structure sheaf as a module over the base and denote by \(\widetilde A\) the fibre product of \(A\) with \(\widetilde B\) over \(B\). Then the inverse image of \(A_\xi\) in \(\widetilde A\) is the inverse image of a locally closed subscheme of \(\widetilde B\) of codimension \(\dim (\underline{\operatorname{Aut}} (\underline X))\) which implies the claim.

In order to calculate the dimension of \(\underline{\operatorname{Aut}} (\underline X)\) it would be preferable to calculate instead the dimension of the group of automorphisms of the Dieudonné space of \(\underline X\). For this we define Dieudonné spaces with PEL-structure over arbitrary schemes which allows us to speak of the moduli space of these Dieudonné spaces \(D\). Using the crystalline Dieudonné functor we get a morphism \(D\): \(B\to D\) which induces a homeomorphism of the underlying Zariski spaces. It follows that the stratification by isomorphism classes of Dieudonné spaces is equal to the Oort stratification. Further we get \(\dim(\underline {\operatorname{Aut}}(\underline X))= \dim (\underline{\text{Aut}} \bigl(D(\underline X)\bigr)\).

We now give an overview of the structure of this work: In the first chapter truncated Barsotti-Tate groups with PEL-structure and their moduli space are defined. In the second chapter we show that the canonical morphism of the deformation functor of an abelian variety (or its Barsotti-Tate group) into the deformation functor of its \(p^n\)-torsion is formally smooth for \(p>2\).

In the third chapter we deduce that the moduli space of truncated Barsotti-Tate groups is smooth over \(\mathbb{Z}_{(p)}\) and that its fibres are of dimension zero. In the fourth chapter we endow the set of isomorphism classes of truncated Barsotti-Tate groups with a topology using the general language of algebraic stacks and in the fifth chapter we relate truncated Barsotti-Tate groups and Dieudonné spaces and define a morphism between their respective moduli spaces and show that this is a homeomorphism. The sixth chapter finally gives the definition of the Oort stratification of a scheme \(S\) with respect to a truncated Barsotti-Tate group with additional structures over \(S\). In particular we define the Oort stratification of the model of a Shimura variety of PEL-type at a prime of good reduction and give the main theorem. After that we give some conditions when the Oort strata are nonempty. In the last chapter we give two examples.

For the entire collection see [Zbl 0958.00023].

##### MSC:

14G35 | Modular and Shimura varieties |

14K10 | Algebraic moduli of abelian varieties, classification |

14D99 | Families, fibrations in algebraic geometry |

14L05 | Formal groups, \(p\)-divisible groups |

14F30 | \(p\)-adic cohomology, crystalline cohomology |

14A20 | Generalizations (algebraic spaces, stacks) |