In contemporary many years, $p$-adic geometry and $p$-adic cohomology theories became quintessential instruments in quantity thought, algebraic geometry, and the speculation of automorphic representations. The Arizona iciness institution 2007, on which the present ebook relies, used to be a distinct chance to introduce graduate scholars to this topic. Following worthwhile introductions by means of John Tate and Vladimir Berkovich, pioneers of non-archimedean geometry, Brian Conrad's bankruptcy introduces the overall thought of Tate's inflexible analytic areas, Raynaud's view of them because the commonplace fibers of formal schemes, and Berkovich areas. Samit Dasgupta and Jeremy Teitelbaum speak about the $p$-adic higher part aircraft for instance of a inflexible analytic house and provides functions to quantity thought (modular types and the $p$-adic Langlands program). Matthew Baker deals an in depth dialogue of the Berkovich projective line and $p$-adic power idea on that and extra normal Berkovich curves. ultimately, Kiran Kedlaya discusses theoretical and computational elements of $p$-adic cohomology and the zeta capabilities of sorts. This publication can be a welcome boost to the library of any graduate pupil and researcher who's attracted to studying concerning the recommendations of $p$-adic geometry.
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