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The moduli spaces of polarized abelian varieties. (English) Zbl 0839.14036
In this article we study the moduli spaces $${\mathcal A}_{g,d}$$ over $$\text{Spec} (\mathbb{Z})$$ parametrizing abelian varieties of dimension $$g$$ with a polarization of degree $$d^2$$. We use the language of algebraic stacks. – Although these moduli spaces are in a sense the most basic moduli spaces of abelian varieties not much seems to be known about their local and global structure. D. Mumford, in Actes Congr. internat. Math. 1970, Vol. 1, 457-465 (1971; Zbl 0222.14023), determined the components of $${\mathcal A}_{g,d}$$ using the spaces $${\mathcal A}_{g, \delta}$$ parametrizing the polarized abelian varieties with a fixed sequence of “elementary divisors” $$\delta=(\delta_1, \dots, \delta_g)$$ of positive integers $$\delta_1 |\delta_2 \dots |\delta_g$$ such that $$\prod^g_{i=1} \delta_i=d$$ associated to the polarization. S. E. Crick jun. [Am. J. Math. 97, 851-861 (1975; Zbl 0321.14020)] determined the local structure of $${\mathcal A}_{g,d}$$ at a point of $${\mathcal A}_{g, \delta}$$. Finally, P. Norman and F. Oort [Ann. Math., II. Ser. 112, 413-439 (1980; Zbl 0483.14010)] studied the stratification by $$p$$-rank $${\mathcal V}_0 \subset {\mathcal V}_1 \subset \cdots \subset {\mathcal V}_{g-1} \subset {\mathcal V}_g={\mathcal A}_{g,d} \otimes \mathbb{F}_p$$; it is shown there that the dimension of $${\mathcal V}_{g-i}$$ is equal to $${1 \over 2} g(g+1)-i$$.
In section 1 we (re)define the locally closed substacks $${\mathcal A}_{g,\delta} \subset {\mathcal A}_{g,d}$$ whose closures are the components of $${\mathcal A}_{g,d}$$. We prove that all geometric fibres of the morphisms $${\mathcal A}_{g, \delta} \to \text{Spec} (\mathbb{Z})$$ are irreducible; this answers a question of Mumford (loc. cit.). It follows that the closures of $${\mathcal A}_{g, \delta} \otimes \overline \mathbb{F}_p$$ are the irreducible components of $${\mathcal A}_{g,d} \otimes \overline \mathbb{F}_p$$. – In section 2 we determine how the fibres of the components of $${\mathcal A}_{g,d}$$ over the finite primes split up into irreducible components. – In section 3 we continue the study of the stratification by $$p$$-rank of $${\mathcal A}_{g,d} \otimes \overline \mathbb{F}_p$$ which was started by P. Norman and F. Oort (loc. cit.).

##### MSC:
 14K10 Algebraic moduli of abelian varieties, classification 14G15 Finite ground fields in algebraic geometry
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##### References:
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