Newton polygon strata in the moduli space of abelian varieties. (English) Zbl 1086.14037

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, 417-440 (2001).
In this fundamental paper the author, after a long period of working, proved the Grothendieck conjecture, a very deep and difficult result concerning the converse of the Grothendieck specialization theorem for \(p\)-divisible groups – the Newton polygon goes up under specialization in a family of \(p\)-divisible groups [A. Grothendieck, Groupes de Barsotti-Tate et cristaux de Dieudonné (Les Presses de l’Universite de Montreal) (1974; Zbl 0331.14021)]. The analogous statement for the reduction mod \(p\) of Siegel moduli spaces is also proved. The proof relies on two rather different aspects of deformation theory of \(p\)-divisible groups. The first one is on deformations of simple \(p\)-divisible groups keeping the Newton polygon constant. The author uses the methods and results derived from “Purity” as obtained in [A. J. de Jong and F. Oort, J. Am. Math. Soc. 13, 209–241 (2000; Zbl 0954.14007)] . The other one is an explicit deformation theory with invariant \(a\)-number one [F. Oort, Ann. Math. (2) 152, 183–206 (2000; Zbl 0991.14016)]. This is an effective method of reading the Newton polygon of a subvariety in a local deformation space. In this paper a combination of these two methods gives the desired result.
For the entire collection see [Zbl 0958.00023].


14K10 Algebraic moduli of abelian varieties, classification
14L05 Formal groups, \(p\)-divisible groups