## Formal subgroups of abelian varieties.(English)Zbl 1064.14047

Summary: We generalize the result of our previous paper [Duke Math. J. 106, No.1, 81–121 (2001; Zbl 1064.14043)] in the following sense. Let $$A$$ be an abelian variety over a number field $$k$$, let $$I$$ be the Néron model of $$A$$ over the ring of integers $$O_k$$ of $$k$$. Completing $$I$$ along its zero section defines a formal group $$\widehat{\mathcal{A}}$$ over $$O_k$$. We prove that any formal subgroup of the generic fiber of $$\widehat{\mathcal{A}}$$ whose closure in $$\widehat{\mathcal{A}}$$ is smooth over an open subset of $$\text{Spec\,}O_k$$ arises in fact from an abelian subvariety of $$A$$. The proof is of a transcendental nature and uses the Arakelovian formalism introduced by J. Bost [Astérisque 237, 115–161 (1996; Zbl 0936.11042)].

### MSC:

 14K15 Arithmetic ground fields for abelian varieties 11G10 Abelian varieties of dimension $$> 1$$ 14K12 Subvarieties of abelian varieties 14L05 Formal groups, $$p$$-divisible groups

### Citations:

Zbl 1064.14043; Zbl 0936.11042
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