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Arithmetische Kompaktifizierung des Modulraums der abelschen Varietäten. (Arithmetical compactification of the moduli space of abelian varieties). (German) Zbl 0597.14036
Arbeitstag. Bonn. 1984, Proc. Meet. Max-Planck-Inst. Math., Bonn 1984, Lect. Notes Math. 1111, 321-383 (1985).
[For the entire collection see Zbl 0547.00007.]
The main object of this article is to construct compactifications of the moduli space of principally polarized abelian varieties of dimension g over Spec(\({\mathbb{Z}})\). The existence of such compactification for each fibre (i.e. Spec(\({\mathbb{Q}})\) etc.) has been known since long. The author succeeded to globalize it in showing that one can apply Mumford’s method of toroidal compactification (over \({\mathbb{C}})\) also in this case by connecting it with another Mumford’s theory [D. Mumford, Compos. Math. 24, 239-272 (1972; Zbl 0241.14020)].
The compactifications are realized as algebraic stacks, but there are finite coverings (over Spec(\({\mathbb{Z}}[1/n]))\) which are algebraic spaces.
As applications, one can simplify some arguments in the proof of Mordell conjecture by the same author [Invent. Math. 73, 349-366 (1983; Zbl 0588.14026)], and one can show the irreducibility of the moduli space of principally polarized abelian varieties over any characteristic (known before only for characteristic 0 or for low dimension).
Reviewer: Y.Namikawa

MSC:
14K10 Algebraic moduli of abelian varieties, classification
14D15 Formal methods and deformations in algebraic geometry
14K15 Arithmetic ground fields for abelian varieties
14D20 Algebraic moduli problems, moduli of vector bundles