The moduli spaces of polarized abelian varieties
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. \textit{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. \textit{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, \textit{P. Norman} and \textit{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.).
scientific article; zbMATH DE number 537220