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Lifting abelian varieties with additional structures. (English) Zbl 1051.14052
The author investigates the moduli space of polarized \({\mathcal O}_B\) abelian varieties with a determinant condition. The main result stated as Theorem 4.5 is that when \(p\) is unramified in \(B\) and greater than 2, the moduli space is a complete intersection relative to the mixed characteristic base in the sense of A. Grothendieck [Eléments de Geométrie Algébrique IV, Publ. Math., Inst. Hautes Étud. Sci. 32, 361 (1967; Zbl 0153.22301), Def. 19.3.6]. The author’s main theorem proves a conjecture of M. Rapoport and Th. Zink [Period spaces for \(p\)-divisible groups, Ann. Math. Stud. 141 (1996; Zbl 0873.14039)] for the case of parahoric level structures which U. Görtz [Adv. Math. 176, 89–115 (2003; Zbl 1051.14027)] has verified for the case of symplectic groups. Section 1 contains the definition of the moduli spaces under consideration with a determinant condition following [R. Kottwitz, J. Am. Math. Soc. 5, 373–444 (1992; Zbl 0796.14014)].
Section 2 reviews the classical classification of Albert of division algebras with a positive involution. Section 3 contains the classification of Dieudonné modules attached to polarized \({\mathcal O}_B\) abelian varieties in terms of normal forms. In section 4, the author uses Grothendieck-Messing’s deformation theory [W. Messing, The crystals associated to Barsotti-Tate groups: with applications to abelian schemes, Lect. Notes Math. 264 (1972; Zbl 0243.14013)] and Fontaine’s theory [J.-M. Fontaine, Groupes p-divisibles sur les corps locaux, Astérisque 47–48 (1977; Zbl 0377.14009)] to investigate the local moduli spaces. The author concludes stating and proving theorem 4.5 relying on Proposition 4.3 and lemma 4.4 contained in this section.

14K05 Algebraic theory of abelian varieties
11G10 Abelian varieties of dimension \(> 1\)
14L05 Formal groups, \(p\)-divisible groups
14K10 Algebraic moduli of abelian varieties, classification
14G20 Local ground fields in algebraic geometry
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