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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.

MSC:
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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