This book presents some of the most important aspects of rigid geometry,
namely its applications to the study of smooth algebraic curves, of
their Jacobians, and of abelian varieties - all of them defined over a
complete non-archimedean valued field. The text starts with a survey of
the foundation of rigid geometry, and then focuses on a detailed
treatment of the applications. In the case of curves with split rational
reduction there is a complete analogue to the fascinating theory of
Riemann surfaces. In the case of proper smooth group varieties the
uniformization and the construction of abelian varieties are treated in
detail.
Rigid geometry was established by John Tate and was enriched by a formal
algebraic approach launched by Michel Raynaud. It has proved as a means
to illustrate the geometric ideas behind the abstract methods of formal
algebraic geometry as used by Mumford and Faltings. This book should be
of great use to students wishing to enter this field, as well as those
already working in it.