Spectral Theory and Analytic Geometry over Non-Archimedean Fields

Vladimir G. Berkovich

08/2012 · Mathematical Surveys and Monographs Livro 33 · American Mathematical Soc.

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The purpose of this book is to introduce a new notion of analytic space
over a non-Archimedean field. Despite the total disconnectedness of
the ground field, these analytic spaces have the usual topological
properties of a complex analytic space, such as local compactness and
local arcwise connectedness. This makes it possible to apply the usual
notions of homotopy and singular homology. The book includes a
homotopic characterization of the analytic spaces associated with
certain classes of algebraic varieties and an interpretation of
Bruhat-Tits buildings in terms of these analytic spaces. The author
also studies the connection with the earlier notion of a rigid analytic
space. Geometrical considerations are used to obtain some
applications, and the analytic spaces are used to construct the
foundations of a non-Archimedean spectral theory of bounded linear
operators. This book requires a background at the level of basic
graduate courses in algebra and topology, as well as some familiarity
with algebraic geometry. It would be of interest to research
mathematicians and graduate students working in algebraic geometry,
number theory, and  -adic analysis.