In contemporary a long time, $p$-adic geometry and $p$-adic cohomology theories became crucial instruments in quantity thought, algebraic geometry, and the idea of automorphic representations. The Arizona wintry weather tuition 2007, on which the present e-book relies, was once a different chance to introduce graduate scholars to this topic. Following worthy introductions by means of John Tate and Vladimir Berkovich, pioneers of non-archimedean geometry, Brian Conrad's bankruptcy introduces the overall conception of Tate's inflexible analytic areas, Raynaud's view of them because the familiar fibers of formal schemes, and Berkovich areas. Samit Dasgupta and Jeremy Teitelbaum talk about the $p$-adic higher part airplane for instance of a inflexible analytic house and provides functions to quantity concept (modular kinds and the $p$-adic Langlands program). Matthew Baker bargains a close dialogue of the Berkovich projective line and $p$-adic power concept on that and extra normal Berkovich curves. eventually, Kiran Kedlaya discusses theoretical and computational points of $p$-adic cohomology and the zeta features of types. This ebook should be a great addition to the library of any graduate pupil and researcher who's attracted to studying concerning the strategies of $p$-adic geometry.

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