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Ordinariness in good reductions of Shimura varieties of PEL-type. (English) Zbl 0983.14024
Let $${\mathbf A}_{g,N}/ \mathbb{P}_p$$ denote the moduli scheme of principally polarized Abelian varieties of a fixed dimension $$g\geq 1$$, over a finite field, $$\mathbb{F}_p$$, with level-$$N$$-structure, where $$N\geq 3$$ is an integer prime to $$p$$, and denote by $$\chi\to {\mathbf A}_{g,N}$$ the universal family. It is a classical result (see below) that the ordinary locus in $${\mathbf A}_{g,N}$$ (that is, the points $$s\in{\mathcal A}_{g,N}$$ such that the $$p$$-divisible group of $$X_{\overline s}$$ has slopes only in $$\{0,1\})$$ is open and dense.
The fact that the ordinary locus is open follows from Grothendieck’s specialization theorem for crystals [A. Grothendieck, “Groupes de Barsotti-Tate et cristaux de Dieudonné” (Montréal 1974; Zbl 0331.14021)].
The fact that the ordinary locus is dense may be proved in three ways:
(a) by deformation theory applied to the stratification of $${\mathbf A}_{g,N}$$ by the $$p$$-rank of $$\chi$$ [N. Koblitz, Compos. Math. 31, 119-218 (1975; Zbl 0332.14008)];
(b) by constructing deformations explicitly [D. Mumford, Algebr. Geom., Bombay Colloq. 1968, 307-322 (1969; Zbl 0216.33101); P. Norman and F. Oort, Ann. Math. (2) 112, 413-439 (1980; Zbl 0483.14010)]; and
(c) by applying Zariski’s connectedness theorem to the construction of a smooth compactification of the moduli stack $${\mathbf A}_g$$ over $$\mathbb{Z}$$ of the principally polarized Abelian varieties [C.-L. Chai and G. Faltings, “Degeneration of abelian varieties” (1990; Zbl 0744.14031)].
In this paper, the author generalizes the ideas of the foregoing to the case of good reduction of Shimura varieties of PEL-type (see below). For such varieties it turns out that the reduction can be considered as a moduli space of Abelian varieties, with additional structures, and in that case a method based on the ideas of method (b) may be used. It is not in general true that the locus is dense when the underlying variety is ordinary.
The author states and proves his principal result in terms of the concept of a $$\mu$$-ordinary locus, which, in the Siegel case, reduces to the ordinary locus. The notion of $$\mu$$-ordinariness depends on that of Shimura varieties in relation to an underlying connected reductive group, $$G$$, and on the notion of the Shimura field, $$E$$, and its $$\nu$$-adic completions at the places $$\nu$$ over primes $$p$$; to each point of the Shimura variety over $$O_{E_\nu}$$ there corresponds an Abelian variety with additional structure. If one takes the arithmetic mean $$\overline\mu\in\overline C$$, for some closed Weyl chamber $$\overline C$$, then the locus of points $$x$$ such that $$\overline\nu(x) =\overline\mu$$ is open and is the $$\mu$$-ordinary locus.
A PEL-datum is a collection of data, $$D$$, related to an algebraic group $$G$$ over $$\mathbb{Q}$$, which is reducible and which is defined in terms of a certain finite dimensional semi-simple $$\mathbb{Q}$$-algebra, equipped with a positive involution, and a finitely generated $$B$$-module, $$V$$, equipped with an alternating, non degenerate $$\mathbb{Q}$$-valued, skew-Hermitian pairing $$\langle\;,\;\rangle$$. One now defines a moduli problem, which turns out to be representable by a moduli scheme $${\mathbf A}_D$$ that is smooth over $$O_E\otimes \mathbb{Z}_{(p)}$$ [R. E. Kottwitz, J. Am. Math. Soc., 5, 373-444, (1992; Zbl 0796.14014)]. In the case when $$G$$ is connected, the generic field of $${\mathbf A}_D$$ consists of isomorphic copies of the Shimura field and the generic fibre of $${\mathbf A}_D$$ consists of isomorphic copies of the Shimura canonical model, $$Sh(G,h)_C$$, over the Shimura field $$E$$. Denote by $$\kappa$$ the residue class field of the $$\nu$$-adic completion $$E_\nu$$ of $$E$$. The author’s main result is the density theorem:
The $$\mu$$-ordinary locus is open and dense in $${\mathbf A}_D\otimes \kappa(O_{E_\nu})$$.
The paper contains a good account of the background and the necessary preliminaries (which the reviewer found particularly helpful) and then deals with the reduction to a deformation problem in four special cases and with the construction of deformations. The author then returns to the four special cases and completes the proof of the main theorem.

##### MSC:
 14K10 Algebraic moduli of abelian varieties, classification 14G35 Modular and Shimura varieties 11G18 Arithmetic aspects of modular and Shimura varieties 14D20 Algebraic moduli problems, moduli of vector bundles
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##### References:
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