You are here  Éléments de Géométrie Rigide, Volume 1  Ahmed
Abbes Publication Date:  Number of Pages:  Progress in
Mathematics 286 [Reviewed by Fernando Q. Gouvêa , on  Classical
algebraic geometry dealt with complex varieties defined by a
system of polynomial equations in several variables. One might,
for example, study the surface in three-dimensional space
defined by the equation x5 + y5 + z5 – 5xyz = 1. In the
classical approach, one thinks of this as a complex manifold
(possibly with some singularities) and studies the resulting
geometry. Complex analysis turns out to be an important tool,
even if we are mostly interested in algebraic questions. To
connect analytic and algebraic questions, there are GAGA
theorems, named after a famous paper of Serre called “Géométrie
Algébrique et Géometrie Analytique” (Annales de l'Institut
Fourier 6: 1–42; there is a Wikipedia page discussing the main
results).  Given the importance and usefulness of p-adic methods
in number theory, it is inevitable that we would like