Formal subgroups of abelian varieties. (English) Zbl 1064.14047

Graftieaux, Philippe

Invent. Math. 145, No. 1, 1-17 (2001).

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