A different subject is treaded in chapter 7, where we give a general treatment of the Schwarz lemma in several variables. This leads to important problems for higher dimensional diophantine investigations.

These Schwarz lemmas are used in chapter 8 for the study of the graph and the image of an analytic homomorphism $\phi : C^{n} \to G_{C}$, in connection with a problem of Weil and Serre on a certain type of characters of the idele - class group of an algebraic number field.

The first appendix, by Daniel Bertrand, gives a survey of the $p$-adic case and of its applications. The second appendix, by Jean-Pierre Serre, provides a proof of several properties of commutatives algebraic groups which are needed in trans­ cendence proofs.