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The moduli spaces of principally polarized abelian varieties with \(\Gamma_ 0(p)\)-level structure. (English) Zbl 0816.14020
Let \((A,\lambda)\) be a p.p. abelian variety of dimension \(g\). A \(\Gamma_ 0 (p)\)-level structure on \((A, \lambda)\) is a flag of subgroup schemes \(0 \subset H_ 1 \subset \cdots \subset H_ g \subset A [p]\) such that \(\;H_ i = p^ i\) and \(H_ g\) is totally isotropic in \(A[p]\). Let \({\mathcal S}(g,p)\) the algebraic stack of finite type over \(\text{Spec} \mathbb{Z}\) classifying p.p. abelian varieties of dimension \(g\) with a \(\Gamma_ 0 (p)\)-level structure. It is obvious that \({\mathcal S}(g,p)\) is smooth over \(\text{Spec} (\mathbb{Z} [1/p])\). The aim of this paper is to study the behaviour of \({\mathcal S}(g,p)\) over the prime \(p\). The main result in this direction is that \({\mathcal S}(2,p)\) is a locally complete intersection morphism and the singularities are not too bad. The ordinary locus is dense in \({\mathcal S}(2,p) \times \text{Spec} (\mathbb{F}_ p)\) and \({\mathcal S} (2,p) \times \text{Spec} (\overline \mathbb{F}_ p)\) has four components.

14K10 Algebraic moduli of abelian varieties, classification
14G20 Local ground fields in algebraic geometry
18G50 Nonabelian homological algebra (category-theoretic aspects)