In recent decades, $p$-adic geometry and $p$-adic cohomology theories
have become indispensable tools in number theory, algebraic geometry,
and the theory of automorphic representations. The Arizona Winter
School 2007, on which the current book is based, was a unique
opportunity to introduce graduate students to this subject. Following
invaluable introductions by John Tate and Vladimir Berkovich, two
pioneers of non-archimedean geometry, Brian Conrad's chapter
introduces the general theory of Tate's rigid analytic spaces,
Raynaud's view of them as the generic fibers of formal schemes, and
Berkovich spaces. Samit Dasgupta and Jeremy Teitelbaum discuss the
$p$-adic upper half plane as an example of a rigid analytic space and
give applications to number theory (modular forms and the $p$-adic
Langlands program). Matthew Baker offers a detailed discussion of the
Berkovich projective line and $p$-adic potential theory on that and
more general Berkovich curves. Finally, Kiran Kedlaya discusses
theoretical and computational aspects of $p$-adic cohomology and the
zeta functions of varieties. This book will be a welcome addition to
the library of any graduate student and researcher who is interested
in learning about the techniques of $p$-adic geometry.
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