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.
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