Appears in collection : 2018 - T1 - WS2 - Model theory and valued fields

The Ax-Lindemann theorem is a functional algebraic independence statement, which is a geometric version of the classical Lindemann-Weierstrass theorem. Itsgeneralizations to uniformizing maps of arithmetic varieties played a key role in recent progress on the Andr´e-Oort conjecture. In this talk I will present a nonarchimedean analogue for the uniformization of products of Mumford curves. In particular, we characterize bi-algebraic irreducible subvarieties of the uniformization. This is joint work with Antoine Chambert-Loir.