The theorey of period mappings has played a central role in nineteen-century mathematics as a fertile place of interaction between algebraic and differential geometry, differential equations, and group theory, from Gauss and Riemann to Klein and Poincaré. This text is an introduction to the p-adic counterpart of this theory, which is much more recent and still mysterious. It should be of interest both to some complex geometers and to some arithmetic geometers.

Starting with an introduction to p-adic analytic geometry (in the sense of Berkovich), it then presents the Rapoport-Zink theory of period mappings, emphasizing the relation with Picard-Fuchs differential equtions. a new theory of fundamental groups, orbifolds, and uniformizing equations (in the p-adic context) accounts for the group-theoretic aspects of these period mappings. The books ends with a theory of p-adictriangle groups. Dr. André's current mathematical interests lie in arithmetic geometry and in the theory of motives.