Arakelov Geometry Over Adelic Curves PDF Download
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Author | : Huayi Chen |
Publisher | : Springer Nature |
Total Pages | : 452 |
Release | : 2020-01-29 |
Genre | : Mathematics |
ISBN | : 9811517282 |
Download Arakelov Geometry over Adelic Curves Book in PDF, ePub and Kindle
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.
Author | : Huayi Chen |
Publisher | : Springer Nature |
Total Pages | : 248 |
Release | : |
Genre | : |
ISBN | : 3031616685 |
Download Positivity in Arakelov Geometry over Adelic Curves Book in PDF, ePub and Kindle
Author | : Huayi Chen |
Publisher | : Birkhäuser |
Total Pages | : 0 |
Release | : 2024-09-08 |
Genre | : Mathematics |
ISBN | : 9783031616679 |
Download Positivity in Arakelov Geometry over Adelic Curves Book in PDF, ePub and Kindle
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.
Author | : Emmanuel Peyre |
Publisher | : Springer Nature |
Total Pages | : 469 |
Release | : 2021-03-10 |
Genre | : Mathematics |
ISBN | : 3030575594 |
Download Arakelov Geometry and Diophantine Applications Book in PDF, ePub and Kindle
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.
Author | : Atsushi Moriwaki |
Publisher | : American Mathematical Soc. |
Total Pages | : 298 |
Release | : 2014-11-05 |
Genre | : Mathematics |
ISBN | : 1470410745 |
Download Arakelov Geometry Book in PDF, ePub and Kindle
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.
Author | : C. Soulé |
Publisher | : Cambridge University Press |
Total Pages | : 190 |
Release | : 1994-09-15 |
Genre | : Mathematics |
ISBN | : 9780521477093 |
Download Lectures on Arakelov Geometry Book in PDF, ePub and Kindle
An account for graduate students of this new technique in diophantine geometry; includes account of higher dimensional theory.
Author | : Hideaki Ikoma |
Publisher | : Cambridge University Press |
Total Pages | : 180 |
Release | : 2022-02-03 |
Genre | : Mathematics |
ISBN | : 1108998194 |
Download The Mordell Conjecture Book in PDF, ePub and Kindle
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.
Author | : Alekos Vidras |
Publisher | : American Mathematical Society |
Total Pages | : 556 |
Release | : 2023-10-18 |
Genre | : Mathematics |
ISBN | : 1470471124 |
Download Multidimensional Residue Theory and Applications Book in PDF, ePub and Kindle
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.
Author | : José Ignacio Burgos Gil |
Publisher | : |
Total Pages | : 0 |
Release | : 2014 |
Genre | : Mathematics |
ISBN | : 9782856297834 |
Download Arithmetic Geometry of Toric Varieties Book in PDF, ePub and Kindle
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.
Author | : Umberto Zannier |
Publisher | : Princeton University Press |
Total Pages | : 175 |
Release | : 2012-03-25 |
Genre | : Mathematics |
ISBN | : 1400842719 |
Download Some Problems of Unlikely Intersections in Arithmetic and Geometry Book in PDF, ePub and Kindle
This book considers the so-called Unlikely Intersections, a topic that embraces well-known issues, such as Lang's and Manin-Mumford's, concerning torsion points in subvarieties of tori or abelian varieties. More generally, the book considers algebraic subgroups that meet a given subvariety in a set of unlikely dimension. The book is an expansion of the Hermann Weyl Lectures delivered by Umberto Zannier at the Institute for Advanced Study in Princeton in May 2010. The book consists of four chapters and seven brief appendixes, the last six by David Masser. The first chapter considers multiplicative algebraic groups, presenting proofs of several developments, ranging from the origins to recent results, and discussing many applications and relations with other contexts. The second chapter considers an analogue in arithmetic and several applications of this. The third chapter introduces a new method for approaching some of these questions, and presents a detailed application of this (by Masser and the author) to a relative case of the Manin-Mumford issue. The fourth chapter focuses on the André-Oort conjecture (outlining work by Pila).