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The geometry of Newton strata in the reduction modulo \(p\) of Shimura varieties of PEL type. (English) Zbl 1335.14008

The main results of the paper under review include a dimension formula for Newton strata and descriptions of their closure of the Newton stratification on the reduction of Shimura varieties of PEL type with hyperspecial level structure.
More specifically, let \(\mathcal{D}\) be a PEL-Shimura datum unramified at a prime \(p>0\) in the sense of R. E. Kottwitz [J. Am. Math. Soc. 5, No. 2, 373–444 (1992; Zbl 0796.14014)] such that the associated linear algebraic group is connected. Let \(\mathcal{A}_0\) be the reduction modulo \(p\) of the associated moduli space defined by Kottwitz. Points of \(\mathcal{A}_0\) correspond to abelian varieties of PEL type. The Newton stratification is the stratification corresponding to the isogeny class of Barsotti-Tate group of points of \(\mathcal{A}_0\). By Dieudonné theory, isogeny classes of Barsotti-Tate groups correspond to a certain finite subset of the \(\sigma\)-conjugacy classes \(B(G_{\mathbb{Q}_p})\) in \(G(\check{\mathbb{Q}}_p)\) where \(\check{\mathbb{Q}}_p\) is the completion of the maximal unramified extension of of \(\mathbb{Q}_p\). A partial order of \(B(G_{\mathbb{Q}_p})\) is given by \(b' \leq b\) if and only if \(b'\) and \(b\) have the same endpoints and that \(b\) lies above \(b'\). For any Newton polygon \(b\), let \(\mathcal{A}_0^b\) be the associated Newton stratum of \(\mathcal{A}_0\). Theorem. The associated Newton stratum \(\mathcal{A}_0^{\leq b}\) is equidimensional of dimension \[ \langle \rho, \mu+\nu(b) \rangle - \frac{1}{2} \text{def}(b), \] where \(\rho\) is the half-sum of absolute positive roots of \(G\), \(\mu\) is the cocharacter induced by \(\mathcal{D}\), and \(\nu(b)\) and \(\text{def}(b)\) denote the Newton point and the defect of \(b\) respectively. The closure of \(\mathcal{A}_0^b\) in \(\mathcal{A}_0\) is \(\mathcal{A}_0^{\leq b}\).
The proof of the main theorem is obtained by calculating the dimension of certain Rapoport-Zink space and it also proves an earlier conjecture of C.-L. Chai [Am. J. Math. 122, No. 5, 967–990 (2000; Zbl 1057.11506)].
The author also gives some applications in deformation theory that generalize conjectures of Grothendieck and Koblitz.
Reviewer: Xiao Xiao (Utica)

MSC:

14G35 Modular and Shimura varieties
14L05 Formal groups, \(p\)-divisible groups
20G25 Linear algebraic groups over local fields and their integers
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References:

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