Spectral Theory and Analytic Geometry over Non-Archimedean Fields (Mathematical Surveys and Monographs)

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 $p$-adic analysis. Rational And Polynomial Parametrizations -- Fractional Linear Transformations -- Cubic Curves -- Cubic Surfaces And General Hypersurfaces -- Outline Of The Theory Of Plane Curves -- Affine Plane And Projective Plane -- Sphere With Handies -- Functions And Differentials On A Curve -- Polynomials And Power Series -- Review Of Abstract Algebra -- Some Commutative Algebra -- Hensel's Lemma And Newton's Theorem -- More About Newton's Theorem -- Branches And Valuations -- Divisors Of Functions And Differentials -- Weierstrass Preparation Theorem -- Intersection Multiplicity -- Resolution Of Singularities Of Plane Curves -- Infinitely Near Singularities -- Parametrizing A Quartic With Three Double Points -- Characteristic Pairs -- Criterion For One Place And Jacobian Problem -- Inversion Formula And Jacobian Problem -- Surfaces -- Hypersurfaces -- Resolution Of Singularities Of Algebraic Surfaces -- Birational And Polyrational Transformations -- Valuations And Birational Correspondence -- Rational Cylinders Through A Variety – Resultants. Shreeram S. Abhyankar. Includes Bibliographical References And Index. Introduces a new notion of analytic space over a non-Archimedean field. 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. Based on lectures presented in courses on algebraic geometry taught by the author at Purdue University, this book covers various topics in the theory of algebraic curves and surfaces, such as rational and polynomial parametrization, functions and differentials on a curve, branches and valuations, and resolution of singularities. Introduces a fresh notion of analytic space over a non-Archimedean field. This 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.