Berkeley Lectures On P Adic Geometry

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Berkeley Lectures on p-adic Geometry

Peter Scholze,Jared Weinstein
Berkeley Lectures on p adic Geometry Book PDF

Author : Peter Scholze,Jared Weinstein
Publisher : Princeton University Press
Page : 264 pages
File Size : 43,9 Mb
Release : 2020-05-26
Category : Mathematics
ISBN : 9780691202150

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Berkeley Lectures on p-adic Geometry by Peter Scholze,Jared Weinstein Pdf

Berkeley Lectures on p-adic Geometry presents an important breakthrough in arithmetic geometry. In 2014, leading mathematician Peter Scholze delivered a series of lectures at the University of California, Berkeley, on new ideas in the theory of p-adic geometry. Building on his discovery of perfectoid spaces, Scholze introduced the concept of “diamonds,” which are to perfectoid spaces what algebraic spaces are to schemes. The introduction of diamonds, along with the development of a mixed-characteristic shtuka, set the stage for a critical advance in the discipline. In this book, Peter Scholze and Jared Weinstein show that the moduli space of mixed-characteristic shtukas is a diamond, raising the possibility of using the cohomology of such spaces to attack the Langlands conjectures for a reductive group over a p-adic field. This book follows the informal style of the original Berkeley lectures, with one chapter per lecture. It explores p-adic and perfectoid spaces before laying out the newer theory of shtukas and their moduli spaces. Points of contact with other threads of the subject, including p-divisible groups, p-adic Hodge theory, and Rapoport-Zink spaces, are thoroughly explained. Berkeley Lectures on p-adic Geometry will be a useful resource for students and scholars working in arithmetic geometry and number theory.

Berkeley Lectures on P-adic Geometry

Peter Scholze,Jared Weinstein
Berkeley Lectures on P adic Geometry Book PDF

Author : Peter Scholze,Jared Weinstein
Publisher : Unknown
Page : 0 pages
File Size : 47,5 Mb
Release : 2021
Category : p-adic analysis
ISBN : OCLC:1396931484

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Berkeley Lectures on P-adic Geometry by Peter Scholze,Jared Weinstein Pdf

This text presents an important breakthrough in arithmetic geometry. In 2014, this work's author delivered a series of lectures at the University of California, Berkeley, on new ideas in the theory of p-adic geometry. Building on his discovery of perfectoid spaces, the author introduced the concept of 'diamonds,' which are to perfectoid spaces what algebraic spaces are to schemes. The introduction of diamonds, along with the development of a mixed-characteristic shtuka, set the stage for a critical advance in the discipline. This book shows that the moduli space of mixed-characteristic shtukas is a diamond, raising the possibility of using the cohomology of such spaces to attack the Langlands conjectures for a reductive group over a p-adic field. The text follows the informal style of the original Berkeley lectures, with one chapter per lecture.

Berkeley Lectures on P-adic Geometry

Peter Scholze,Jared Weinstein
Berkeley Lectures on P adic Geometry Book PDF

Author : Peter Scholze,Jared Weinstein
Publisher : Princeton University Press
Page : 260 pages
File Size : 44,5 Mb
Release : 2020-05-26
Category : Mathematics
ISBN : 9780691202099

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Berkeley Lectures on P-adic Geometry by Peter Scholze,Jared Weinstein Pdf

Berkeley Lectures on p-adic Geometry presents an important breakthrough in arithmetic geometry. In 2014, leading mathematician Peter Scholze delivered a series of lectures at the University of California, Berkeley, on new ideas in the theory of p-adic geometry. Building on his discovery of perfectoid spaces, Scholze introduced the concept of “diamonds,” which are to perfectoid spaces what algebraic spaces are to schemes. The introduction of diamonds, along with the development of a mixed-characteristic shtuka, set the stage for a critical advance in the discipline. In this book, Peter Scholze and Jared Weinstein show that the moduli space of mixed-characteristic shtukas is a diamond, raising the possibility of using the cohomology of such spaces to attack the Langlands conjectures for a reductive group over a p-adic field. This book follows the informal style of the original Berkeley lectures, with one chapter per lecture. It explores p-adic and perfectoid spaces before laying out the newer theory of shtukas and their moduli spaces. Points of contact with other threads of the subject, including p-divisible groups, p-adic Hodge theory, and Rapoport-Zink spaces, are thoroughly explained. Berkeley Lectures on p-adic Geometry will be a useful resource for students and scholars working in arithmetic geometry and number theory.

p-adic Differential Equations

Kiran S. Kedlaya
p adic Differential Equations Book PDF

Author : Kiran S. Kedlaya
Publisher : Cambridge University Press
Page : 518 pages
File Size : 55,9 Mb
Release : 2022-06-09
Category : Mathematics
ISBN : 9781009275651

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p-adic Differential Equations by Kiran S. Kedlaya Pdf

Now in its second edition, this volume provides a uniquely detailed study of $P$-adic differential equations. Assuming only a graduate-level background in number theory, the text builds the theory from first principles all the way to the frontiers of current research, highlighting analogies and links with the classical theory of ordinary differential equations. The author includes many original results which play a key role in the study of $P$-adic geometry, crystalline cohomology, $P$-adic Hodge theory, perfectoid spaces, and algorithms for L-functions of arithmetic varieties. This updated edition contains five new chapters, which revisit the theory of convergence of solutions of $P$-adic differential equations from a more global viewpoint, introducing the Berkovich analytification of the projective line, defining convergence polygons as functions on the projective line, and deriving a global index theorem in terms of the Laplacian of the convergence polygon.

p-adic Hodge Theory

Bhargav Bhatt,Martin Olsson
p adic Hodge Theory Book PDF

Author : Bhargav Bhatt,Martin Olsson
Publisher : Springer Nature
Page : 325 pages
File Size : 45,9 Mb
Release : 2020-06-15
Category : Mathematics
ISBN : 9783030438449

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p-adic Hodge Theory by Bhargav Bhatt,Martin Olsson Pdf

This proceedings volume contains articles related to the research presented at the 2017 Simons Symposium on p-adic Hodge theory. This symposium was focused on recent developments in p-adic Hodge theory, especially those concerning integral questions and their connections to notions in algebraic topology. This volume features original research articles as well as articles that contain new research and survey some of these recent developments. It is the first of three volumes dedicated to p-adic Hodge theory.

Supersingular P-adic L-functions, Maass-Shimura Operators and Waldspurger Formulas

Daniel Kriz
Supersingular P adic L functions Maass Shimura Operators and Waldspurger Formulas Book PDF

Author : Daniel Kriz
Publisher : Princeton University Press
Page : 280 pages
File Size : 51,7 Mb
Release : 2021-11-09
Category : Mathematics
ISBN : 9780691216478

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Supersingular P-adic L-functions, Maass-Shimura Operators and Waldspurger Formulas by Daniel Kriz Pdf

A groundbreaking contribution to number theory that unifies classical and modern results This book develops a new theory of p-adic modular forms on modular curves, extending Katz's classical theory to the supersingular locus. The main novelty is to move to infinite level and extend coefficients to period sheaves coming from relative p-adic Hodge theory. This makes it possible to trivialize the Hodge bundle on the infinite-level modular curve by a "canonical differential" that restricts to the Katz canonical differential on the ordinary Igusa tower. Daniel Kriz defines generalized p-adic modular forms as sections of relative period sheaves transforming under the Galois group of the modular curve by weight characters. He introduces the fundamental de Rham period, measuring the position of the Hodge filtration in relative de Rham cohomology. This period can be viewed as a counterpart to Scholze's Hodge-Tate period, and the two periods satisfy a Legendre-type relation. Using these periods, Kriz constructs splittings of the Hodge filtration on the infinite-level modular curve, defining p-adic Maass-Shimura operators that act on generalized p-adic modular forms as weight-raising operators. Through analysis of the p-adic properties of these Maass-Shimura operators, he constructs new p-adic L-functions interpolating central critical Rankin-Selberg L-values, giving analogues of the p-adic L-functions of Katz, Bertolini-Darmon-Prasanna, and Liu-Zhang-Zhang for imaginary quadratic fields in which p is inert or ramified. These p-adic L-functions yield new p-adic Waldspurger formulas at special values.

Perfectoid Spaces

Bhargav Bhatt,Ana Caraiani,Kiran S. Kedlaya,Peter Scholze,Jared Weinstein
Perfectoid Spaces Book PDF

Author : Bhargav Bhatt,Ana Caraiani,Kiran S. Kedlaya,Peter Scholze,Jared Weinstein
Publisher : American Mathematical Society
Page : 297 pages
File Size : 44,9 Mb
Release : 2022-02-04
Category : Mathematics
ISBN : 9781470465100

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Perfectoid Spaces by Bhargav Bhatt,Ana Caraiani,Kiran S. Kedlaya,Peter Scholze,Jared Weinstein Pdf

Introduced by Peter Scholze in 2011, perfectoid spaces are a bridge between geometry in characteristic 0 and characteristic $p$, and have been used to solve many important problems, including cases of the weight-monodromy conjecture and the association of Galois representations to torsion classes in cohomology. In recognition of the transformative impact perfectoid spaces have had on the field of arithmetic geometry, Scholze was awarded a Fields Medal in 2018. This book, originating from a series of lectures given at the 2017 Arizona Winter School on perfectoid spaces, provides a broad introduction to the subject. After an introduction with insight into the history and future of the subject by Peter Scholze, Jared Weinstein gives a user-friendly and utilitarian account of the theory of adic spaces. Kiran Kedlaya further develops the foundational material, studies vector bundles on Fargues–Fontaine curves, and introduces diamonds and shtukas over them with a view toward the local Langlands correspondence. Bhargav Bhatt explains the application of perfectoid spaces to comparison isomorphisms in $p$-adic Hodge theory. Finally, Ana Caraiani explains the application of perfectoid spaces to the construction of Galois representations associated to torsion classes in the cohomology of locally symmetric spaces for the general linear group. This book will be an invaluable asset for any graduate student or researcher interested in the theory of perfectoid spaces and their applications.

Arithmetic and Geometry over Local Fields

Bruno Anglès,Tuan Ngo Dac
Arithmetic and Geometry over Local Fields Book PDF

Author : Bruno Anglès,Tuan Ngo Dac
Publisher : Springer Nature
Page : 337 pages
File Size : 54,5 Mb
Release : 2021-03-03
Category : Mathematics
ISBN : 9783030662493

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Arithmetic and Geometry over Local Fields by Bruno Anglès,Tuan Ngo Dac Pdf

This volume introduces some recent developments in Arithmetic Geometry over local fields. Its seven chapters are centered around two common themes: the study of Drinfeld modules and non-Archimedean analytic geometry. The notes grew out of lectures held during the research program "Arithmetic and geometry of local and global fields" which took place at the Vietnam Institute of Advanced Study in Mathematics (VIASM) from June to August 2018. The authors, leading experts in the field, have put great effort into making the text as self-contained as possible, introducing the basic tools of the subject. The numerous concrete examples and suggested research problems will enable graduate students and young researchers to quickly reach the frontiers of this fascinating branch of mathematics.

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 : 48,5 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.

Algebraic Geometry

Richard Thomas
Algebraic Geometry Book PDF

Author : Richard Thomas
Publisher : American Mathematical Soc.
Page : 635 pages
File Size : 54,6 Mb
Release : 2018-06-01
Category : Geometry, Algebraic
ISBN : 9781470435783

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Algebraic Geometry by Richard Thomas Pdf

This is Part 2 of a two-volume set. Since Oscar Zariski organized a meeting in 1954, there has been a major algebraic geometry meeting every decade: Woods Hole (1964), Arcata (1974), Bowdoin (1985), Santa Cruz (1995), and Seattle (2005). The American Mathematical Society has supported these summer institutes for over 50 years. Their proceedings volumes have been extremely influential, summarizing the state of algebraic geometry at the time and pointing to future developments. The most recent Summer Institute in Algebraic Geometry was held July 2015 at the University of Utah in Salt Lake City, sponsored by the AMS with the collaboration of the Clay Mathematics Institute. This volume includes surveys growing out of plenary lectures and seminar talks during the meeting. Some present a broad overview of their topics, while others develop a distinctive perspective on an emerging topic. Topics span both complex algebraic geometry and arithmetic questions, specifically, analytic techniques, enumerative geometry, moduli theory, derived categories, birational geometry, tropical geometry, Diophantine questions, geometric representation theory, characteristic and -adic tools, etc. The resulting articles will be important references in these areas for years to come.

Lectures on Logarithmic Algebraic Geometry

Arthur Ogus
Lectures on Logarithmic Algebraic Geometry Book PDF

Author : Arthur Ogus
Publisher : Cambridge University Press
Page : 559 pages
File Size : 53,5 Mb
Release : 2018-11-08
Category : Mathematics
ISBN : 9781107187733

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Lectures on Logarithmic Algebraic Geometry by Arthur Ogus Pdf

A self-contained introduction to logarithmic geometry, a key tool for analyzing compactification and degeneration 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 : 40,6 Mb
Release : 2008
Category : Mathematics
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.

Lectures on the Geometry of Quantization

Sean Bates,Alan Weinstein
Lectures on the Geometry of Quantization Book PDF

Author : Sean Bates,Alan Weinstein
Publisher : American Mathematical Soc.
Page : 150 pages
File Size : 42,8 Mb
Release : 1997
Category : Mathematics
ISBN : 0821807986

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Lectures on the Geometry of Quantization by Sean Bates,Alan Weinstein Pdf

These notes are based on a course entitled ``Symplectic Geometry and Geometric Quantization'' taught by Alan Weinstein at the University of California, Berkeley (fall 1992) and at the Centre Emile Borel (spring 1994). The only prerequisite for the course needed is a knowledge of the basic notions from the theory of differentiable manifolds (differential forms, vector fields, transversality, etc.). The aim is to give students an introduction to the ideas of microlocal analysis and the related symplectic geometry, with an emphasis on the role these ideas play in formalizing the transition between the mathematics of classical dynamics (hamiltonian flows on symplectic manifolds) and quantum mechanics (unitary flows on Hilbert spaces). These notes are meant to function as a guide to the literature. The authors refer to other sources for many details that are omitted and can be bypassed on a first reading.

Perfectoid Spaces: Lectures from the 2017 Arizona Winter School

Bryden Cais,Bhargav Bhatt,Ana Caraiani,Kiran S. Kedlaya,Peter Scholze,Jared Weinstein
Perfectoid Spaces Lectures from the 2017 Arizona Winter School Book PDF

Author : Bryden Cais,Bhargav Bhatt,Ana Caraiani,Kiran S. Kedlaya,Peter Scholze,Jared Weinstein
Publisher : American Mathematical Soc.
Page : 297 pages
File Size : 44,5 Mb
Release : 2019-10-01
Category : Topological fields
ISBN : 9781470450151

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Perfectoid Spaces: Lectures from the 2017 Arizona Winter School by Bryden Cais,Bhargav Bhatt,Ana Caraiani,Kiran S. Kedlaya,Peter Scholze,Jared Weinstein Pdf

Introduced by Peter Scholze in 2011, perfectoid spaces are a bridge between geometry in characteristic 0 and characteristic p, and have been used to solve many important problems, including cases of the weight-monodromy conjecture and the association of Galois representations to torsion classes in cohomology. In recognition of the transformative impact perfectoid spaces have had on the field of arithmetic geometry, Scholze was awarded a Fields Medal in 2018. This book, originating from a series of lectures given at the 2017 Arizona Winter School on perfectoid spaces, provides a broad introduction to the subject. After an introduction with insight into the history and future of the subject by Peter Scholze, Jared Weinstein gives a user-friendly and utilitarian account of the theory of adic spaces. Kiran Kedlaya further develops the foundational material, studies vector bundles on Fargues–Fontaine curves, and introduces diamonds and shtukas over them with a view toward the local Langlands correspondence. Bhargav Bhatt explains the application of perfectoid spaces to comparison isomorphisms in p-adic Hodge theory. Finally, Ana Caraiani explains the application of perfectoid spaces to the construction of Galois representations associated to torsion classes in the cohomology of locally symmetric spaces for the general linear group. This book will be an invaluable asset for any graduate student or researcher interested in the theory of perfectoid spaces and their applications.

Proceedings Of The International Congress Of Mathematicians 2018 (Icm 2018) (In 4 Volumes)

Sirakov Boyan,Souza Paulo Ney De,Viana Marcelo
Proceedings Of The International Congress Of Mathematicians 2018 Icm 2018 In 4 Volumes Book PDF

Author : Sirakov Boyan,Souza Paulo Ney De,Viana Marcelo
Publisher : World Scientific
Page : 5396 pages
File Size : 49,6 Mb
Release : 2019-02-27
Category : Mathematics
ISBN : 9789813272897

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Proceedings Of The International Congress Of Mathematicians 2018 (Icm 2018) (In 4 Volumes) by Sirakov Boyan,Souza Paulo Ney De,Viana Marcelo Pdf

The Proceedings of the ICM publishes the talks, by invited speakers, at the conference organized by the International Mathematical Union every 4 years. It covers several areas of Mathematics and it includes the Fields Medal and Nevanlinna, Gauss and Leelavati Prizes and the Chern Medal laudatios.