Arakelov Geometry Over Adelic Curves
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Arakelov Geometry Over Adelic Curves
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Author : Huayi Chen
language : en
Publisher: Springer Nature
Release Date : 2020-01-29
Arakelov Geometry Over Adelic Curves written by Huayi Chen and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020-01-29 with Mathematics categories.
The purpose of this book is to build the fundament of an Arakelov theory over adelic curves in order to provide a unified framework for research on arithmetic geometry in several directions. By adelic curve is meant a field equipped with a family of absolute values parametrized by a measure space, such that the logarithmic absolute value of each non-zero element of the field is an integrable function on the measure space. In the literature, such construction has been discussed in various settings which are apparently transversal to each other. The authors first formalize the notion of adelic curves and discuss in a systematic way its algebraic covers, which are important in the study of height theory of algebraic points beyond Weil–Lang’s height theory. They then establish a theory of adelic vector bundles on adelic curves, which considerably generalizes the classic geometry of vector bundles or that of Hermitian vector bundles over an arithmetic curve. They focus on an analogue of the slope theory in the setting of adelic curves and in particular estimate the minimal slope of tensor product adelic vector bundles. Finally, by using the adelic vector bundles as a tool, a birational Arakelov geometry for projective variety over an adelic curve is developed. As an application, a vast generalization of Nakai–Moishezon’s criterion of positivity is proven in clarifying the arguments of geometric nature from several fundamental results in the classic geometry of numbers. Assuming basic knowledge of algebraic geometry and algebraic number theory, the book is almost self-contained. It is suitable for researchers in arithmetic geometry as well as graduate students focusing on these topics for their doctoral theses.
Positivity In Arakelov Geometry Over Adelic Curves
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Author : Huayi Chen
language : en
Publisher: Springer Nature
Release Date : 2024-08-21
Positivity In Arakelov Geometry Over Adelic Curves written by Huayi Chen and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-08-21 with Mathematics categories.
This monograph presents new research on Arakelov geometry over adelic curves, a novel theory of arithmetic geometry developed by the authors. It explores positivity conditions and establishes the Hilbert-Samuel formula and the equidistribution theorem in the context of adelic curves. Connections with several classical topics in Arakelov geometry and Diophantine geometry are highlighted, such as the arithmetic Hilbert-Samuel formula, positivity of line bundles, equidistribution of small subvarieties, and theorems resembling the Bogomolov conjecture. Detailed proofs and explanations are provided to ensure the text is accessible to both graduate students and experienced researchers.
Arakelov Geometry And Diophantine Applications
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Author : Emmanuel Peyre
language : en
Publisher: Springer Nature
Release Date : 2021-03-10
Arakelov Geometry And Diophantine Applications written by Emmanuel Peyre and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-03-10 with Mathematics categories.
Bridging the gap between novice and expert, the aim of this book is to present in a self-contained way a number of striking examples of current diophantine problems to which Arakelov geometry has been or may be applied. Arakelov geometry can be seen as a link between algebraic geometry and diophantine geometry. Based on lectures from a summer school for graduate students, this volume consists of 12 different chapters, each written by a different author. The first chapters provide some background and introduction to the subject. These are followed by a presentation of different applications to arithmetic geometry. The final part describes the recent application of Arakelov geometry to Shimura varieties and the proof of an averaged version of Colmez's conjecture. This book thus blends initiation to fundamental tools of Arakelov geometry with original material corresponding to current research. This book will be particularly useful for graduate students and researchers interested in the connections between algebraic geometry and number theory. The prerequisites are some knowledge of number theory and algebraic geometry.
Arakelov Geometry
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Author : Atsushi Moriwaki
language : en
Publisher: American Mathematical Soc.
Release Date : 2014-11-05
Arakelov Geometry written by Atsushi Moriwaki and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014-11-05 with Mathematics categories.
The main goal of this book is to present the so-called birational Arakelov geometry, which can be viewed as an arithmetic analog of the classical birational geometry, i.e., the study of big linear series on algebraic varieties. After explaining classical results about the geometry of numbers, the author starts with Arakelov geometry for arithmetic curves, and continues with Arakelov geometry of arithmetic surfaces and higher-dimensional varieties. The book includes such fundamental results as arithmetic Hilbert-Samuel formula, arithmetic Nakai-Moishezon criterion, arithmetic Bogomolov inequality, the existence of small sections, the continuity of arithmetic volume function, the Lang-Bogomolov conjecture and so on. In addition, the author presents, with full details, the proof of Faltings' Riemann-Roch theorem. Prerequisites for reading this book are the basic results of algebraic geometry and the language of schemes.
Multidimensional Residue Theory And Applications
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Author : Alekos Vidras
language : en
Publisher: American Mathematical Society
Release Date : 2023-10-18
Multidimensional Residue Theory And Applications written by Alekos Vidras and has been published by American Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-10-18 with Mathematics categories.
Residue theory is an active area of complex analysis with connections and applications to fields as diverse as partial differential and integral equations, computer algebra, arithmetic or diophantine geometry, and mathematical physics. Multidimensional Residue Theory and Applications defines and studies multidimensional residues via analytic continuation for holomorphic bundle-valued current maps. This point of view offers versatility and flexibility to the tools and constructions proposed, allowing these residues to be defined and studied outside the classical case of complete intersection. The book goes on to show how these residues are algebraic in nature, and how they relate and apply to a wide range of situations, most notably to membership problems, such as the Briançon–Skoda theorem and Hilbert's Nullstellensatz, to arithmetic intersection theory and to tropical geometry. This book will supersede the existing literature in this area, which dates back more than three decades. It will be appreciated by mathematicians and graduate students in multivariate complex analysis. But thanks to the gentle treatment of the one-dimensional case in Chapter 1 and the rich background material in the appendices, it may also be read by specialists in arithmetic, diophantine, or tropical geometry, as well as in mathematical physics or computer algebra.
Arithmetic Geometry Of Toric Varieties
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Author : José Ignacio Burgos Gil
language : en
Publisher:
Release Date : 2014
Arithmetic Geometry Of Toric Varieties written by José Ignacio Burgos Gil and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2014 with Mathematics categories.
The authors show that the height of a toric variety with respect to a toric metrized line bundle can be expressed as the integral over a polytope of a certain adelic family of concave functions. To state and prove this result, the authors study the Arakelov geometry of toric varieties. In particular, they consider models over a discrete valuation ring, metrized line bundles, and their associated measures and heights. They show that these notions can be translated in terms of convex analysis and are closely related to objects such as polyhedral complexes, concave functions, real Monge-Ampere measures, and Legendre-Fenchel duality. The authors also present a closed formula for the integral over a polytope of a function of one variable composed with a linear form. This formula allows them to compute the height of toric varieties with respect to some interesting metrics arising from polytopes and compute the height of toric projective curves with respect to the Fubini-Study metric and the height of some toric bundles.
The Mordell Conjecture
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Author : Atsushi Moriwaki
language : en
Publisher:
Release Date : 2022
The Mordell Conjecture written by Atsushi Moriwaki and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022 with MATHEMATICS categories.
"The Mordell conjecture (Faltings's theorem) is one of the most important achievements in Diophantine geometry, stating that an algebraic curve of genus at least two has only finitely many rational points. This book provides a self-contained and detailed proof of the Mordell conjecture following the papers of Bombieri and Vojta. Also acting as a concise introduction to Diophantine geometry, the text starts from basics of algebraic number theory, touches on several important theorems and techniques (including the theory of heights, the Mordell- Weil theorem, Siegel's lemma and Roth's lemma) from Diophantine geometry, and culminates in the proof of the Mordell conjecture. Based on the authors' own teaching experience, it will be of great value to advanced undergraduate and graduate students in algebraic geometry and number theory, as well as researchers interested in Diophantine geometry as a whole"--
Lectures On Arakelov Geometry
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Author : C. Soulé
language : en
Publisher: Cambridge University Press
Release Date : 1994-09-15
Lectures On Arakelov Geometry written by C. Soulé and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1994-09-15 with Mathematics categories.
An account for graduate students of this new technique in diophantine geometry; includes account of higher dimensional theory.
Beyond Primes
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Author : N.B. Singh
language : en
Publisher: N.B. Singh
Release Date :
Beyond Primes written by N.B. Singh and has been published by N.B. Singh this book supported file pdf, txt, epub, kindle and other format this book has been release on with Mathematics categories.
"Beyond Primes" delves into the fascinating world of number theory beyond the realm of prime numbers. From exploring topics like composite numbers, perfect numbers, and cryptographically significant numbers, to investigating unsolved problems and conjectures in number theory, this book offers readers a captivating journey into the depths of mathematical exploration. With clear explanations and intriguing examples, "Beyond Primes" is an essential read for anyone interested in the beauty and complexity of number theory, offering insights into the mysteries that lie beyond the realm of primes.
Foundations Of Arithmetic Differential Geometry
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Author : Alexandru Buium
language : en
Publisher: American Mathematical Soc.
Release Date : 2017-06-09
Foundations Of Arithmetic Differential Geometry written by Alexandru Buium and has been published by American Mathematical Soc. this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-06-09 with Mathematics categories.
The aim of this book is to introduce and develop an arithmetic analogue of classical differential geometry. In this new geometry the ring of integers plays the role of a ring of functions on an infinite dimensional manifold. The role of coordinate functions on this manifold is played by the prime numbers. The role of partial derivatives of functions with respect to the coordinates is played by the Fermat quotients of integers with respect to the primes. The role of metrics is played by symmetric matrices with integer coefficients. The role of connections (respectively curvature) attached to metrics is played by certain adelic (respectively global) objects attached to the corresponding matrices. One of the main conclusions of the theory is that the spectrum of the integers is “intrinsically curved”; the study of this curvature is then the main task of the theory. The book follows, and builds upon, a series of recent research papers. A significant part of the material has never been published before.