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Serre-Tate theory for moduli spaces of PEL type. (English) Zbl 1107.11028
In this profound paper the author generalizes the Serre-Tate theory to the ordinary locus of good reduction of Shimura varieties of PEL-type. Let \(\mathcal D\) be a usual PEL datum and let \(\mathcal A_{\mathcal D}\) be the moduli space of abelian varieties with the prescribed additional structures. Assume that \(p>2\) and the moduli space \(\mathcal A_{\mathcal D}\) has good reduction at \(p\). On the reduction \(\mathcal A_{\mathcal D}\otimes \overline {\mathbb F}_p\), there are the Newton polygon stratification and the Ekedahl-Oort stratification. The first one comes from the classification of the associated \(p\)-divisible groups with additional structures up to isogeny, which is intensively studied by F. Oort [Faber, Carel (ed.) et al., Moduli of abelian varieties. Proceedings of the 3rd Texel conference, Texel Island, Netherlands, April 1999. Basel: Birkhäuser. Prog. Math. 195, 417–440 (2001; Zbl 1086.14037)], R. E. Kottwitz [Compos. Math. 56, 201–220 (1985; Zbl 0597.20038)], M. Rapoport and M. Richartz [Compos. Math. 103, 153–181 (1996; Zbl 0874.14008)], C.-L. Chai [Am. J. Math. 122, 967–990 (2000; Zbl 1057.11506)], and many others. The latter stratification comes from the classification of the associated \(\text{ BT}_1\)’s with additional structures up to isomorphism. This is studied by F. Oort [Faber, Carel (ed.) et al., Moduli of abelian varieties. Proceedings of the 3rd Texel conference, Texel Island, Netherlands, April 1999. Basel: Birkhäuser. Prog. Math. 195, 345–416 (2001; Zbl 1052.14047)], E. Z. Goren and F. Oort [J. Algebr. Geom. 9, 111–154 (2000; Zbl 0973.14010)], T. Wedhorn [Faber, Carel (ed.) et al., Moduli of abelian varieties. Proceedings of the 3rd Texel conference, Texel Island, Netherlands, April 1999. Basel: Birkhäuser. Prog. Math. 195, 441–471 (2001; Zbl 1052.14026)], and the author [Faber, Carel (ed.) et al., Moduli of abelian varieties. Proceedings of the 3rd Texel conference, Texel Island, Netherlands, April 1999. Basel: Birkhäuser. Prog. Math. 195, 255–298 (2001; Zbl 1084.14523)].
A point in \(\mathcal A_{\mathcal D}\otimes \overline {\mathbb F}_p\) is said to be \(\mu\)-ordinary if it lies in an open NP stratum. A point in \(\mathcal A_{\mathcal D}\otimes \overline {\mathbb F}_p\) is said to be \([p]\)-ordinary if it lies an open EO stratum. The first main result the author proves is that these two notions of ordinary agree. Using another result of the author and T. Wedhorn on the dimensions of the EO strata [math.Ag/0208161, to appear in Ann. Inst. Fourier(Grenoble)], [loc. cit. Zbl 1052.14026)], the author gives another proof of the density of the \(\mu\)-ordinary locus, which is proved by T. Wedhorn [Ann. Sci. École Norm. Sp. (4) 32, No. 5, 575–618 (1999; Zbl 0983.14024)].
The Serre-Tate theory says that the formal deformation space of an ordinary abelian variety has a natural structure of formal torus. The main result of this paper under review is to generalize the Serre-Tate theory to the ordinary locus of the considered moduli space. The author proves that the formal deformation of an ordinary object has a “group-like” structure called cascade, which is a generalization of the notion of a biextension. Roughly speaking, it is a sequence of tree-like fibration, which is the extension part of graded pieces of the slope filtration under forgetful maps, with fibers close to a B.-T. group. Recently, instead of looking at the whole formal deformation, Chai considered subvarieties with a fixed isomorphism type of the associated \(p\)-divisible groups, called leaves. This fine geometric structure as introduced by Oort. Chai proved that any formal completion of a leaf in the Siegel moduli space has similar group-like fibration structure. This further investigation provides the satisfactory generalized Serre-Tate theory.
In the last section the author gives a very interesting application on congruence relations of the Frobenius correspondences on the ordinary locus.

MSC:
11G18 Arithmetic aspects of modular and Shimura varieties
14K05 Algebraic theory of abelian varieties
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