Rigid Analytic Geometry And Its Applications

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Rigid Analytic Geometry and Its Applications

Jean Fresnel,Marius van der Put
Rigid Analytic Geometry and Its Applications Book PDF

Author : Jean Fresnel,Marius van der Put
Publisher : Springer Science & Business Media
Page : 303 pages
File Size : 41,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461200413

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Rigid Analytic Geometry and Its Applications by Jean Fresnel,Marius van der Put Pdf

Rigid (analytic) spaces were invented to describe degenerations, reductions, and moduli of algebraic curves and abelian varieties. This work, a revised and greatly expanded new English edition of an earlier French text by the same authors, presents important new developments and applications of the theory of rigid analytic spaces to abelian varieties, "points of rigid spaces," étale cohomology, Drinfeld modular curves, and Monsky-Washnitzer cohomology. The exposition is concise, self-contained, rich in examples and exercises, and will serve as an excellent graduate-level text for the classroom or for self-study.

Foundations of Rigid Geometry I

Kazuhiro Fujiwara,Fumiharu Kato
Foundations of Rigid Geometry I Book PDF

Author : Kazuhiro Fujiwara,Fumiharu Kato
Publisher : Unknown
Page : 863 pages
File Size : 43,5 Mb
Release : 2018
Category : MATHEMATICS
ISBN : 3037196351

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Foundations of Rigid Geometry I by Kazuhiro Fujiwara,Fumiharu Kato Pdf

Rigid geometry is one of the modern branches of algebraic and arithmetic geometry. It has its historical origin in J. Tate's rigid analytic geometry, which aimed at developing an analytic geometry over non-archimedean valued fields. Nowadays, rigid geometry is a discipline in its own right and has acquired vast and rich structures, based on discoveries of its relationship with birational and formal geometries. In this research monograph, foundational aspects of rigid geometry are discussed, putting emphasis on birational and topological features of rigid spaces. Besides the rigid geometry itself, topics include the general theory of formal schemes and formal algebraic spaces, based on a theory of complete rings which are not necessarily Noetherian. Also included is a discussion on the relationship with Tate's original rigid analytic geometry, V.G. Berkovich's analytic geometry and R. Huber's adic spaces. As a model example of applications, a proof of Nagata's compactification theorem for schemes is given in the appendix. The book is encyclopedic and almost self-contained.

Lectures on Formal and Rigid Geometry

Siegfried Bosch
Lectures on Formal and Rigid Geometry Book PDF

Author : Siegfried Bosch
Publisher : Springer
Page : 254 pages
File Size : 51,9 Mb
Release : 2014-08-22
Category : Mathematics
ISBN : 9783319044170

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Lectures on Formal and Rigid Geometry by Siegfried Bosch Pdf

The aim of this work is to offer a concise and self-contained 'lecture-style' introduction to the theory of classical rigid geometry established by John Tate, together with the formal algebraic geometry approach launched by Michel Raynaud. These Lectures are now viewed commonly as an ideal means of learning advanced rigid geometry, regardless of the reader's level of background. Despite its parsimonious style, the presentation illustrates a number of key facts even more extensively than any other previous work. This Lecture Notes Volume is a revised and slightly expanded version of a preprint that appeared in 2005 at the University of Münster's Collaborative Research Center "Geometrical Structures in Mathematics".

Rigid Geometry of Curves and Their Jacobians

Werner Lütkebohmert
Rigid Geometry of Curves and Their Jacobians Book PDF

Author : Werner Lütkebohmert
Publisher : Springer
Page : 386 pages
File Size : 52,7 Mb
Release : 2016-01-26
Category : Mathematics
ISBN : 9783319273716

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Rigid Geometry of Curves and Their Jacobians by Werner Lütkebohmert Pdf

This book presents some of the most important aspects of rigid geometry, namely its applications to the study of smooth algebraic curves, of their Jacobians, and of abelian varieties - all of them defined over a complete non-archimedean valued field. The text starts with a survey of the foundation of rigid geometry, and then focuses on a detailed treatment of the applications. In the case of curves with split rational reduction there is a complete analogue to the fascinating theory of Riemann surfaces. In the case of proper smooth group varieties the uniformization and the construction of abelian varieties are treated in detail. Rigid geometry was established by John Tate and was enriched by a formal algebraic approach launched by Michel Raynaud. It has proved as a means to illustrate the geometric ideas behind the abstract methods of formal algebraic geometry as used by Mumford and Faltings. This book should be of great use to students wishing to enter this field, as well as those already working in it.

Spectral Theory and Analytic Geometry over Non-Archimedean Fields

Vladimir G. Berkovich
Spectral Theory and Analytic Geometry over Non Archimedean Fields Book PDF

Author : Vladimir G. Berkovich
Publisher : American Mathematical Soc.
Page : 169 pages
File Size : 55,5 Mb
Release : 2012-08-02
Category : Algebraic number theory
ISBN : 9780821890202

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Spectral Theory and Analytic Geometry over Non-Archimedean Fields by Vladimir G. Berkovich Pdf

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.

Berkovich Spaces and Applications

Antoine Ducros,Charles Favre,Johannes Nicaise
Berkovich Spaces and Applications Book PDF

Author : Antoine Ducros,Charles Favre,Johannes Nicaise
Publisher : Springer
Page : 413 pages
File Size : 41,5 Mb
Release : 2014-11-21
Category : Mathematics
ISBN : 9783319110295

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Berkovich Spaces and Applications by Antoine Ducros,Charles Favre,Johannes Nicaise Pdf

We present an introduction to Berkovich’s theory of non-archimedean analytic spaces that emphasizes its applications in various fields. The first part contains surveys of a foundational nature, including an introduction to Berkovich analytic spaces by M. Temkin, and to étale cohomology by A. Ducros, as well as a short note by C. Favre on the topology of some Berkovich spaces. The second part focuses on applications to geometry. A second text by A. Ducros contains a new proof of the fact that the higher direct images of a coherent sheaf under a proper map are coherent, and B. Rémy, A. Thuillier and A. Werner provide an overview of their work on the compactification of Bruhat-Tits buildings using Berkovich analytic geometry. The third and final part explores the relationship between non-archimedean geometry and dynamics. A contribution by M. Jonsson contains a thorough discussion of non-archimedean dynamical systems in dimension 1 and 2. Finally a survey by J.-P. Otal gives an account of Morgan-Shalen's theory of compactification of character varieties. This book will provide the reader with enough material on the basic concepts and constructions related to Berkovich spaces to move on to more advanced research articles on the subject. We also hope that the applications presented here will inspire the reader to discover new settings where these beautiful and intricate objects might arise.

Valuation Theory and Its Applications

Franz-Viktor Kuhlmann,Salma Kuhlmann,Murray Marshall
Valuation Theory and Its Applications Book PDF

Author : Franz-Viktor Kuhlmann,Salma Kuhlmann,Murray Marshall
Publisher : American Mathematical Soc.
Page : 470 pages
File Size : 44,5 Mb
Release : 2002-01-01
Category : Mathematics
ISBN : 0821871390

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Valuation Theory and Its Applications by Franz-Viktor Kuhlmann,Salma Kuhlmann,Murray Marshall Pdf

This book is the first of two proceedings volumes stemming from the International Conference and Workshop on Valuation Theory held at the University of Saskatchewan (Saskatoon, SK, Canada). Valuation theory arose in the early part of the twentieth century in connection with number theory and has many important applications to geometry and analysis: the classical application to the study of algebraic curves and to Dedekind and Prufer domains; the close connection to the famousresolution of the singularities problem; the study of the absolute Galois group of a field; the connection between ordering, valuations, and quadratic forms over a formally real field; the application to real algebraic geometry; the study of noncommutative rings; etc. The special feature of this book isits focus on current applications of valuation theory to this broad range of topics. Also included is a paper on the history of valuation theory. The book is suitable for graduate students and research mathematicians working in algebra, algebraic geometry, number theory, and mathematical logic.

Snowbird Lectures in Algebraic Geometry

Ravi Vakil,American Mathematical Society
Snowbird Lectures in Algebraic Geometry Book PDF

Author : Ravi Vakil,American Mathematical Society
Publisher : American Mathematical Soc.
Page : 202 pages
File Size : 44,5 Mb
Release : 2005
Category : Geometry, Algebraic
ISBN : 9780821837191

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Snowbird Lectures in Algebraic Geometry by Ravi Vakil,American Mathematical Society Pdf

A significant part of the 2004 Summer Research Conference on Algebraic Geometry (Snowbird, UT) was devoted to lectures introducing the participants, in particular, graduate students and recent Ph.D.'s, to a wide swathe of algebraic geometry and giving them a working familiarity with exciting, rapidly developing parts of the field. One of the main goals of the organizers was to allow the participants to broaden their horizons beyond the narrow area in which they are working. A fine selection of topics and a noteworthy list of contributors made the resulting collection of articles a useful resource for everyone interested in getting acquainted with the modern topic of algebraic geometry. The book consists of ten articles covering, among others, the following topics: the minimal model program, derived categories of sheaves on algebraic varieties, Kobayashi hyperbolicity, groupoids and quotients in algebraic geometry, rigid analytic varieties, and equivariant cohomology. Suitable for independent study, this unique volume is intended for graduate students and researchers interested in algebraic geometry.

$p$-adic Geometry

Matthew Baker
p adic Geometry Book PDF

Author : Matthew Baker
Publisher : American Mathematical Soc.
Page : 220 pages
File Size : 49,8 Mb
Release : 2008
Category : Arithmetical algebraic geometry
ISBN : 9780821844687

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$p$-adic Geometry by Matthew Baker Pdf

"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 Schoo1 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."--BOOK JACKET.

Rigid Cohomology over Laurent Series Fields

Christopher Lazda,Ambrus Pál
Rigid Cohomology over Laurent Series Fields Book PDF

Author : Christopher Lazda,Ambrus Pál
Publisher : Springer
Page : 267 pages
File Size : 54,5 Mb
Release : 2016-04-27
Category : Mathematics
ISBN : 9783319309514

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Rigid Cohomology over Laurent Series Fields by Christopher Lazda,Ambrus Pál Pdf

In this monograph, the authors develop a new theory of p-adic cohomology for varieties over Laurent series fields in positive characteristic, based on Berthelot's theory of rigid cohomology. Many major fundamental properties of these cohomology groups are proven, such as finite dimensionality and cohomological descent, as well as interpretations in terms of Monsky-Washnitzer cohomology and Le Stum's overconvergent site. Applications of this new theory to arithmetic questions, such as l-independence and the weight monodromy conjecture, are also discussed. The construction of these cohomology groups, analogous to the Galois representations associated to varieties over local fields in mixed characteristic, fills a major gap in the study of arithmetic cohomology theories over function fields. By extending the scope of existing methods, the results presented here also serve as a first step towards a more general theory of p-adic cohomology over non-perfect ground fields. Rigid Cohomology over Laurent Series Fields will provide a useful tool for anyone interested in the arithmetic of varieties over local fields of positive characteristic. Appendices on important background material such as rigid cohomology and adic spaces make it as self-contained as possible, and an ideal starting point for graduate students looking to explore aspects of the classical theory of rigid cohomology and with an eye towards future research in the subject.

Perfectoid Spaces

Debargha Banerjee,Kiran S. Kedlaya,Ehud de Shalit,Chitrabhanu Chaudhuri
Perfectoid Spaces Book PDF

Author : Debargha Banerjee,Kiran S. Kedlaya,Ehud de Shalit,Chitrabhanu Chaudhuri
Publisher : Springer Nature
Page : 395 pages
File Size : 52,7 Mb
Release : 2022-04-21
Category : Mathematics
ISBN : 9789811671210

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Perfectoid Spaces by Debargha Banerjee,Kiran S. Kedlaya,Ehud de Shalit,Chitrabhanu Chaudhuri Pdf

This book contains selected chapters on perfectoid spaces, their introduction and applications, as invented by Peter Scholze in his Fields Medal winning work. These contributions are presented at the conference on “Perfectoid Spaces” held at the International Centre for Theoretical Sciences, Bengaluru, India, from 9–20 September 2019. The objective of the book is to give an advanced introduction to Scholze’s theory and understand the relation between perfectoid spaces and some aspects of arithmetic of modular (or, more generally, automorphic) forms such as representations mod p, lifting of modular forms, completed cohomology, local Langlands program, and special values of L-functions. All chapters are contributed by experts in the area of arithmetic geometry that will facilitate future research in the direction.

Étale Cohomology of Rigid Analytic Varieties and Adic Spaces

Roland Huber
tale Cohomology of Rigid Analytic Varieties and Adic Spaces Book PDF

Author : Roland Huber
Publisher : Springer
Page : 450 pages
File Size : 54,6 Mb
Release : 2013-07-01
Category : Mathematics
ISBN : 9783663099918

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Étale Cohomology of Rigid Analytic Varieties and Adic Spaces by Roland Huber Pdf

Diese Forschungsmonographie von hohem mathematischen Niveau liefert einen neuen Zugang zu den rigid-analytischen Räumen, sowie ihrer etalen Kohomologie.USP: Aus der Froschung: Zahlentheorie und Algebraische Geometrie

Parabolic Quasilinear Equations Minimizing Linear Growth Functionals

Fuensanta Andreu-Vaillo,Vicent Caselles,José M. Mazon
Parabolic Quasilinear Equations Minimizing Linear Growth Functionals Book PDF

Author : Fuensanta Andreu-Vaillo,Vicent Caselles,José M. Mazon
Publisher : Birkhäuser
Page : 342 pages
File Size : 42,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034879286

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Parabolic Quasilinear Equations Minimizing Linear Growth Functionals by Fuensanta Andreu-Vaillo,Vicent Caselles,José M. Mazon Pdf

This book details the mathematical developments in total variation based image restauration. From the reviews: "This book is devoted to PDE's of elliptic and parabolic type associated to functionals having a linear growth in the gradient, with a special emphasis on the applications related to image restoration and nonlinear filters....The book is written with great care, paying also a lot of attention to the bibliographical and historical notes."-- ZENTRALBLATT MATH

Algebraic and Combinatorial Aspects of Tropical Geometry

Erwan Brugalle,Maria Angelica Cueto,Alicia Dickenstein,Eva-Maria Feichtner,Ilia Itenberg
Algebraic and Combinatorial Aspects of Tropical Geometry Book PDF

Author : Erwan Brugalle,Maria Angelica Cueto,Alicia Dickenstein,Eva-Maria Feichtner,Ilia Itenberg
Publisher : American Mathematical Soc.
Page : 363 pages
File Size : 46,8 Mb
Release : 2013-05-23
Category : Mathematics
ISBN : 9780821891469

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Algebraic and Combinatorial Aspects of Tropical Geometry by Erwan Brugalle,Maria Angelica Cueto,Alicia Dickenstein,Eva-Maria Feichtner,Ilia Itenberg Pdf

This volume contains the proceedings of the CIEM workshop on Tropical Geometry, held December 12-16, 2011, at the International Centre for Mathematical Meetings (CIEM), Castro Urdiales, Spain. Tropical geometry is a new and rapidly developing field of mat

Complex Convexity and Analytic Functionals

Mats Andersson,Mikael Passare,Ragnar Sigurdsson
Complex Convexity and Analytic Functionals Book PDF

Author : Mats Andersson,Mikael Passare,Ragnar Sigurdsson
Publisher : Birkhäuser
Page : 164 pages
File Size : 44,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783034878715

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Complex Convexity and Analytic Functionals by Mats Andersson,Mikael Passare,Ragnar Sigurdsson Pdf

This book puts the modern theory of complex linear convexity on a solid footing, and gives a thorough and up-to-date survey of its current status. Applications include the Fantappié transformation of analytic functionals, integral representation formulas, polynomial interpolation, and solutions to linear partial differential equations.