On the Newton stratification. (English) Zbl 1159.14304

Séminaire Bourbaki. Volume 2001/2002. Exposés 894–908. Paris: Société Mathématique de France (ISBN 2-85629-149-X/pbk). Astérisque 290, 207-224, Exp. No. 903 (2003).
This Bourbaki report is a clear, concise survey of a constellation of results concerning the Newton stratification induced by a family of isocrystals over a base scheme, with special attention to the Siegel moduli space.
Briefly, the first section recalls the definitions of \(F\)-(iso)crystals and their relation to \(p\)-divisible groups, including the partial ordering of Newton polygons and Grothendieck’s semicontinuity theorem.
The second section sketches the ideas behind the purity theorem of A. J. de Jong and F. Oort [J. Am. Math. Soc. 13, No. 1, 209–241 (2000; Zbl 0954.14007)]. Roughly speaking, this theorem states that if a Newton stratification is nonconstant, then the jumps in the stratification occur in codimension one.
The third section reviews some of Oort’s work on the Newton stratification of the Siegel moduli space [F. Oort, Progr. Math. 195, 417–440 (2001; Zbl 1086.14037)]. Oort calculates the dimension of (each component of) each Newton stratum, and shows that the closure of each Newton stratum is a union of Newton strata. In addition to the purity theorem mentioned above, Oort’s proof makes vital use of deformation theory.
The next section summarizes work of T. Zink on displays [in: Cohomologies \(p\)-adiques et applications arithmétiques (I). Paris: Société Mathématique de France. Astérisque. 278, 127–248 (2002; Zbl 1008.14008)], which provides an efficient method for explicitly computing deformations of a \(p\)-divisible group, and thus of an abelian variety in positive characteristic.
The report closes with generalizations, some still conjectural, to other moduli spaces of PEL type.
For the entire collection see [Zbl 1050.00006].


14K10 Algebraic moduli of abelian varieties, classification
11G10 Abelian varieties of dimension \(> 1\)
11G18 Arithmetic aspects of modular and Shimura varieties
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