Uniformisation des variétés abéliennes

Abstract

An abelian variety Z over a complete valued field k, which has a bad reduction, can be uniformized in the category of formal schemes (or rigid analytic spaces) over the valuation ring of k. This means Z ≃ G/Λ where G is an algebraic group, namely an extension of an abelian variety with good reduction by a torus of rank h, and where Λ ≃ Z^h is a discrete subgroup of G. In the proof one reduces to the case where Z = the Jacobian variety of a curve C. The construction of G and Λ uses line bundles on Ω, the universal covering of C in the category of formal schemes over the valuation ring of k.