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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.

##### MSC:
 14K10 Algebraic moduli of abelian varieties, classification 14G20 Local ground fields in algebraic geometry 18G50 Nonabelian homological algebra (category-theoretic aspects)
##### Keywords:
abelian variety; algebraic stack; complete intersection