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Linearity properties of Shimura varieties. II. (English) Zbl 0960.14012

For Part I, cf. J. Algebr. Geom. 7, No. 3, 539-567 (1998; Zbl 0956.14016).
Let \({\mathfrak A}\) be the moduli scheme over \(\mathbb Z [1/n]\) of principally polarized abelian varieties with level \(n\) structure, whose generic fiber can be described as a Shimura variety. Given an ordinary moduli point \(x \in {\mathfrak A}\) in characteristic \(p\), a local linearity property at \(x\) can be formulated in terms of the Serre-Tate group structure on the formal deformation space, that is, the formal completion of \({\mathfrak A}\) at \(x\). In the case of characteristic zero, such a property is closely linked to a differential-geometric linearity property. In this paper the author proves that an irreducible algebraic subvariety of \({\mathfrak A}\) is a Shimura subvariety if, locally at an ordinary point \(x\), it is formally linear. As an application, he reformulates Oort’s conjecture on subvarieties \(Z \hookrightarrow {\mathfrak A}\) with a dense collection of CM-points and proves it in a special case.

MSC:

14G35 Modular and Shimura varieties
14K12 Subvarieties of abelian varieties
14D07 Variation of Hodge structures (algebro-geometric aspects)
11G18 Arithmetic aspects of modular and Shimura varieties

Citations:

Zbl 0956.14016
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