Rational Points On Algebraic Varieties

Author by : Emmanuel Peyre
Languange : en
Publisher by : Birkhäuser
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Description : This book is devoted to the study of rational and integral points on higher-dimensional algebraic varieties. It contains carefully selected research papers addressing the arithmetic geometry of varieties which are not of general type, with an emphasis on how rational points are distributed with respect to the classical, Zariski and adelic topologies. The present volume gives a glimpse of the state of the art of this rapidly expanding domain in arithmetic geometry. The techniques involve explicit geometric constructions, ideas from the minimal model program in algebraic geometry as well as analytic number theory and harmonic analysis on adelic groups.


Brauer Groups Tamagawa Measures And Rational Points On Algebraic Varieties

Author by : Jorg Jahnel
Languange : en
Publisher by : American Mathematical Soc.
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Description : The central theme of this book is the study of rational points on algebraic varieties of Fano and intermediate type--both in terms of when such points exist and, if they do, their quantitative density. The book consists of three parts. In the first part, the author discusses the concept of a height and formulates Manin's conjecture on the asymptotics of rational points on Fano varieties. The second part introduces the various versions of the Brauer group. The author explains why a Brauer class may serve as an obstruction to weak approximation or even to the Hasse principle. This part includes two sections devoted to explicit computations of the Brauer-Manin obstruction for particular types of cubic surfaces. The final part describes numerical experiments related to the Manin conjecture that were carried out by the author together with Andreas-Stephan Elsenhans. The book presents the state of the art in computational arithmetic geometry for higher-dimensional algebraic varieties and will be a valuable reference for researchers and graduate students interested in that area.


Higher Dimensional Varieties And Rational Points

Author by : Károly Jr. Böröczky
Languange : en
Publisher by : Springer Science & Business Media
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Description : Exploring the connections between arithmetic and geometric properties of algebraic varieties has been the object of much fruitful study for a long time, especially in the case of curves. The aim of the Summer School and Conference on "Higher Dimensional Varieties and Rational Points" held in Budapest, Hungary during September 2001 was to bring together students and experts from the arithmetic and geometric sides of algebraic geometry in order to get a better understanding of the current problems, interactions and advances in higher dimension. The lecture series and conference lectures assembled in this volume give a comprehensive introduction to students and researchers in algebraic geometry and in related fields to the main ideas of this rapidly developing area.


Many Rational Points

Author by : N.E. Hurt
Languange : en
Publisher by : Springer Science & Business Media
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Description : This volume provides a source book of examples with relationships to advanced topics regarding Sato-Tate conjectures, Eichler-Selberg trace formula, Katz-Sarnak conjectures and Hecke operators." "The book will be of use to mathematicians, physicists and engineers interested in the mathematical methods of algebraic geometry as they apply to coding theory and cryptography."--Jacket.


Some Results On Binary Forms And Counting Rational Points On Algebraic Varieties

Author by : Stanley Yao Xiao
Languange : en
Publisher by : Unknown
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Description : In this thesis we study several problems related to the representation of integers by binary forms and counting rational points on algebraic varieties. In particular, we establish an asymptotic formula for $R_F(Z)$, the number of integers of absolute value up to $Z$ which can be represented by a binary form $F$ with integer coefficients, degree $d \geq 3$, and non-zero discriminant. We give superior results when $d = 3$ or $4$, which completely resolves the cases considered by Hooley. We establish an asymptotic formula for the number of pairs $(x,y) \in \bZ^2$ such that $F(x,y)$ is $k$-free, whenever $F$ satisfies certain necessary conditions and $k > 7d/18$. Finally, we give various results on the arithmetic of certain cubic and quartic surfaces as well as general methods to estimate the number of rational points of bounded height on algebraic varieties. In particular, we give a bound for the density of rational points on del Pezzo surfaces of degree $2$. These results depend on generalizations of Salberger's global determinant method in various settings.


Birational Geometry Rational Curves And Arithmetic

Author by : Fedor Bogomolov
Languange : en
Publisher by : Springer Science & Business Media
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Total Read : 52
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Description : ​​​​This book features recent developments in a rapidly growing area at the interface of higher-dimensional birational geometry and arithmetic geometry. It focuses on the geometry of spaces of rational curves, with an emphasis on applications to arithmetic questions. Classically, arithmetic is the study of rational or integral solutions of diophantine equations and geometry is the study of lines and conics. From the modern standpoint, arithmetic is the study of rational and integral points on algebraic varieties over nonclosed fields. A major insight of the 20th century was that arithmetic properties of an algebraic variety are tightly linked to the geometry of rational curves on the variety and how they vary in families. This collection of solicited survey and research papers is intended to serve as an introduction for graduate students and researchers interested in entering the field, and as a source of reference for experts working on related problems. Topics that will be addressed include: birational properties such as rationality, unirationality, and rational connectedness, existence of rational curves in prescribed homology classes, cones of rational curves on rationally connected and Calabi-Yau varieties, as well as related questions within the framework of the Minimal Model Program.


Rational Points Rational Curves And Entire Holomorphic Curves On Projective Varieties

Author by : Carlo Gasbarri
Languange : en
Publisher by : American Mathematical Soc.
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Description : This volume contains papers from the Short Thematic Program on Rational Points, Rational Curves, and Entire Holomorphic Curves and Algebraic Varieties, held from June 3-28, 2013, at the Centre de Recherches Mathématiques, Université de Montréal, Québec, Canada. The program was dedicated to the study of subtle interconnections between geometric and arithmetic properties of higher-dimensional algebraic varieties. The main areas of the program were, among others, proving density of rational points in Zariski or analytic topology on special varieties, understanding global geometric properties of rationally connected varieties, as well as connections between geometry and algebraic dynamics exploring new geometric techniques in Diophantine approximation. This book is co-published with the Centre de Recherches Mathématiques.


Integral Points On Algebraic Varieties

Author by : Pietro Corvaja
Languange : en
Publisher by : Springer
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Total Read : 75
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Description : This book is intended to be an introduction to Diophantine geometry. The central theme of the book is to investigate the distribution of integral points on algebraic varieties. This text rapidly introduces problems in Diophantine geometry, especially those involving integral points, assuming a geometrical perspective. It presents recent results not available in textbooks and also new viewpoints on classical material. In some instances, proofs have been replaced by a detailed analysis of particular cases, referring to the quoted papers for complete proofs. A central role is played by Siegel’s finiteness theorem for integral points on curves. The book ends with the analysis of integral points on surfaces.


Rational Approximations On Smooth Rational Surfaces

Author by : Diana Carolina Castañeda Santos
Languange : en
Publisher by : Unknown
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Description : In this thesis, we study a conjecture made by D. McKinnon about rational approximations to rational points in algebraic varieties. The conjecture states that if a rational point P on a variety X lies on a rational curve, then the best approximations to P can be chosen to lie along a rational curve on X. According to the conditions of the conjecture, it is natural to study this problem on algebraic varieties that contain a dense subset of rational points. Motivated by this remark, we study the conjecture on smooth rational surfaces, which not only contain a dense set of rational points, but also their classification is well understood. Given a point P on an algebraic variety and an ample divisor D, the approximation constant measures how well P can be approximated by rational points on the variety, with respect to a height function associated to D. In the study of the conjecture, it became clear that if a curve C contains the best approximations to P with respect to ample divisors D and D', then C turns out to be also a curve containing the best approximations for any divisor that is a linear combination of D and D'. This property motivated the study of the nef cone of the algebraic variety. Every ample divisor belongs to the interior of the nef cone and can be written as a linear combination of the generators of the nef cone. By an exhaustive study of the effective and nef cones on a smooth rational surface, it is possible to find a curve that contains the best approximations to the point P with respect to an ample divisor, which can be written in terms of the generators of the nef cone. In this work we use the fact that a smooth rational surface is obtained by a finite number of blow-ups of a Hirzebruch surface or of the projective plane. The Hirzebruch surfaces are equipped with morphisms to the projective line and to cones in some projective space. The study of the fibres of these morphisms provide good candidates of curves with best approximations and we rely on them to prove the conjecture for these cases. We review the conjecture proved by McKinnon in the case of smooth rational surfaces of Picard rank 4. We explore some of the examples in this case to present the techniques using the nef cone of the variety, and then we extend the result for surfaces of bigger Picard ranks. Finally, we extend the result to surfaces obtained by blowing up an arbitrary number of times on smooth points of the reducible fibre of the map to the projective line.


Arithmetic And Geometry

Author by : Luis Dieulefait
Languange : en
Publisher by : Cambridge University Press
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Total Read : 54
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Description : The 'Arithmetic and Geometry' trimester, held at the Hausdorff Research Institute for Mathematics in Bonn, focussed on recent work on Serre's conjecture and on rational points on algebraic varieties. The resulting proceedings volume provides a modern overview of the subject for graduate students in arithmetic geometry and Diophantine geometry. It is also essential reading for any researcher wishing to keep abreast of the latest developments in the field. Highlights include Tim Browning's survey on applications of the circle method to rational points on algebraic varieties and Per Salberger's chapter on rational points on cubic hypersurfaces.


Arithmetic Geometry

Author by : Clay Mathematics Institute. Summer School
Languange : en
Publisher by : American Mathematical Soc.
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Description : This book is based on survey lectures given at the 2006 Clay Summer School on Arithmetic Geometry at the Mathematics Institute of the University of Gottingen. Intended for graduate students and recent Ph.D.'s, this volume will introduce readers to modern techniques and outstanding conjectures at the interface of number theory and algebraic geometry. The main focus is rational points on algebraic varieties over non-algebraically closed fields. Do they exist? If not, can this be proven efficiently and algorithmically? When rational points do exist, are they finite in number and can they be found effectively? When there are infinitely many rational points, how are they distributed? For curves, a cohesive theory addressing these questions has emerged in the last few decades. Highlights include Faltings' finiteness theorem and Wiles's proof of Fermat's Last Theorem. Key techniques are drawn from the theory of elliptic curves, including modular curves and parametrizations, Heegner points, and heights. The arithmetic of higher-dimensional varieties is equally rich, offering a complex interplay of techniques including Shimura varieties, the minimal model program, moduli spaces of curves and maps, deformation theory, Galois cohomology, harmonic analysis, and automorphic functions. However, many foundational questions about the structure of rational points remain open, and research tends to focus on properties of specific classes of varieties.


Encyclopedic Dictionary Of Mathematics

Author by : Mathematical Society of Japan
Languange : en
Publisher by : MIT Press
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Description : V.1. A.N. v.2. O.Z. Apendices and indexes.


Rational Points And Arithmetic Of Fundamental Groups

Author by : Jakob Stix
Languange : en
Publisher by : Springer
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Total Read : 26
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Description : The section conjecture in anabelian geometry, announced by Grothendieck in 1983, is concerned with a description of the set of rational points of a hyperbolic algebraic curve over a number field in terms of the arithmetic of its fundamental group. While the conjecture is still open today in 2012, its study has revealed interesting arithmetic for curves and opened connections, for example, to the question whether the Brauer-Manin obstruction is the only one against rational points on curves. This monograph begins by laying the foundations for the space of sections of the fundamental group extension of an algebraic variety. Then, arithmetic assumptions on the base field are imposed and the local-to-global approach is studied in detail. The monograph concludes by discussing analogues of the section conjecture created by varying the base field or the type of variety, or by using a characteristic quotient or its birational analogue in lieu of the fundamental group extension.


Torsors Tale Homotopy And Applications To Rational Points

Author by : Alexei N. Skorobogatov
Languange : en
Publisher by : Cambridge University Press
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Description : Torsors, also known as principal bundles or principal homogeneous spaces, are ubiquitous in mathematics. The purpose of this book is to present expository lecture notes and cutting-edge research papers on the theory and applications of torsors and étale homotopy, all written from different perspectives by leading experts. Part one of the book contains lecture notes on recent uses of torsors in geometric invariant theory and representation theory, plus an introduction to the étale homotopy theory of Artin and Mazur. Part two of the book features a milestone paper on the étale homotopy approach to the arithmetic of rational points. Furthermore, the reader will find a collection of research articles on algebraic groups and homogeneous spaces, rational and K3 surfaces, geometric invariant theory, rational points, descent and the Brauer–Manin obstruction. Together, these give a state-of-the-art view of a broad area at the crossroads of number theory and algebraic geometry.


Cohomological And Geometric Approaches To Rationality Problems

Author by : Fedor Bogomolov
Languange : en
Publisher by : Springer Science & Business Media
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Total Read : 56
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Description : Rationality problems link algebra to geometry, and the difficulties involved depend on the transcendence degree of $K$ over $k$, or geometrically, on the dimension of the variety. A major success in 19th century algebraic geometry was a complete solution of the rationality problem in dimensions one and two over algebraically closed ground fields of characteristic zero. Such advances has led to many interdisciplinary applications to algebraic geometry. This comprehensive book consists of surveys of research papers by leading specialists in the field and gives indications for future research in rationality problems. Topics discussed include the rationality of quotient spaces, cohomological invariants of quasi-simple Lie type groups, rationality of the moduli space of curves, and rational points on algebraic varieties. This volume is intended for researchers, mathematicians, and graduate students interested in algebraic geometry, and specifically in rationality problems. Contributors: F. Bogomolov; T. Petrov; Y. Tschinkel; Ch. Böhning; G. Catanese; I. Cheltsov; J. Park; N. Hoffmann; S. J. Hu; M. C. Kang; L. Katzarkov; Y. Prokhorov; A. Pukhlikov


Number Theory And Algebraic Geometry

Author by : Miles Reid
Languange : en
Publisher by : Cambridge University Press
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Description : This volume honors Sir Peter Swinnerton-Dyer's mathematical career spanning more than 60 years' of amazing creativity in number theory and algebraic geometry.


Rational Points On Varieties

Author by : Bjorn Poonen
Languange : en
Publisher by : American Mathematical Soc.
Format Available : PDF, ePub, Mobi
Total Read : 50
Total Download : 335
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Description : This book is motivated by the problem of determining the set of rational points on a variety, but its true goal is to equip readers with a broad range of tools essential for current research in algebraic geometry and number theory. The book is unconventional in that it provides concise accounts of many topics instead of a comprehensive account of just one—this is intentionally designed to bring readers up to speed rapidly. Among the topics included are Brauer groups, faithfully flat descent, algebraic groups, torsors, étale and fppf cohomology, the Weil conjectures, and the Brauer-Manin and descent obstructions. A final chapter applies all these to study the arithmetic of surfaces. The down-to-earth explanations and the over 100 exercises make the book suitable for use as a graduate-level textbook, but even experts will appreciate having a single source covering many aspects of geometry over an unrestricted ground field and containing some material that cannot be found elsewhere.


Arithmetic Of Higher Dimensional Algebraic Varieties

Author by : Bjorn Poonen
Languange : en
Publisher by : Springer Science & Business Media
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Total Read : 90
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Description : This text offers a collection of survey and research papers by leading specialists in the field documenting the current understanding of higher dimensional varieties. Recently, it has become clear that ideas from many branches of mathematics can be successfully employed in the study of rational and integral points. This book will be very valuable for researchers from these various fields who have an interest in arithmetic applications, specialists in arithmetic geometry itself, and graduate students wishing to pursue research in this area.


Homotopy Theory And Arithmetic Geometry Motivic And Diophantine Aspects

Author by : Frank Neumann
Languange : en
Publisher by : Springer
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Total Read : 72
Total Download : 903
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Description : This book provides an introduction to state-of-the-art applications of homotopy theory to arithmetic geometry. The contributions to this volume are based on original lectures by leading researchers at the LMS-CMI Research School on ‘Homotopy Theory and Arithmetic Geometry - Motivic and Diophantine Aspects’ and the Nelder Fellow Lecturer Series, which both took place at Imperial College London in the summer of 2018. The contribution by Brazelton, based on the lectures by Wickelgren, provides an introduction to arithmetic enumerative geometry, the notes of Cisinski present motivic sheaves and new cohomological methods for intersection theory, and Schlank’s contribution gives an overview of the use of étale homotopy theory for obstructions to the existence of rational points on algebraic varieties. Finally, the article by Asok and Østvær, based in part on the Nelder Fellow lecture series by Østvær, gives a survey of the interplay between motivic homotopy theory and affine algebraic geometry, with a focus on contractible algebraic varieties. Now a major trend in arithmetic geometry, this volume offers a detailed guide to the fascinating circle of recent applications of homotopy theory to number theory. It will be invaluable to research students entering the field, as well as postdoctoral and more established researchers.


Higher Dimensional Geometry Over Finite Fields

Author by : D. Kaledin
Languange : en
Publisher by : IOS Press
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Total Read : 59
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Description : Number systems based on a finite collection of symbols, such as the 0s and 1s of computer circuitry, are ubiquitous in the modern age. Finite fields are the most important such number systems, playing a vital role in military and civilian communications through coding theory and cryptography. These disciplines have evolved over recent decades, and where once the focus was on algebraic curves over finite fields, recent developments have revealed the increasing importance of higher-dimensional algebraic varieties over finite fields. The papers included in this publication introduce the reader to recent developments in algebraic geometry over finite fields with particular attention to applications of geometric techniques to the study of rational points on varieties over finite fields of dimension of at least 2.


The Brauer Grothendieck Group

Author by : Jean-Louis Colliot-Thélène
Languange : en
Publisher by : Springer Nature
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Description : This monograph provides a systematic treatment of the Brauer group of schemes, from the foundational work of Grothendieck to recent applications in arithmetic and algebraic geometry. The importance of the cohomological Brauer group for applications to Diophantine equations and algebraic geometry was discovered soon after this group was introduced by Grothendieck. The Brauer–Manin obstruction plays a crucial role in the study of rational points on varieties over global fields. The birational invariance of the Brauer group was recently used in a novel way to establish the irrationality of many new classes of algebraic varieties. The book covers the vast theory underpinning these and other applications. Intended as an introduction to cohomological methods in algebraic geometry, most of the book is accessible to readers with a knowledge of algebra, algebraic geometry and algebraic number theory at graduate level. Much of the more advanced material is not readily available in book form elsewhere; notably, de Jong’s proof of Gabber’s theorem, the specialisation method and applications of the Brauer group to rationality questions, an in-depth study of the Brauer–Manin obstruction, and proof of the finiteness theorem for the Brauer group of abelian varieties and K3 surfaces over finitely generated fields. The book surveys recent work but also gives detailed proofs of basic theorems, maintaining a balance between general theory and concrete examples. Over half a century after Grothendieck's foundational seminars on the topic, The Brauer–Grothendieck Group is a treatise that fills a longstanding gap in the literature, providing researchers, including research students, with a valuable reference on a central object of algebraic and arithmetic geometry.


Introduction To Commutative Algebra And Algebraic Geometry

Author by : Ernst Kunz
Languange : en
Publisher by : Springer Science & Business Media
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Description : Originally published in 1985, this classic textbook is an English translation of Einführung in die kommutative Algebra und algebraische Geometrie. As part of the Modern Birkhäuser Classics series, the publisher is proud to make Introduction to Commutative Algebra and Algebraic Geometry available to a wider audience. Aimed at students who have taken a basic course in algebra, the goal of the text is to present important results concerning the representation of algebraic varieties as intersections of the least possible number of hypersurfaces and—a closely related problem—with the most economical generation of ideals in Noetherian rings. Along the way, one encounters many basic concepts of commutative algebra and algebraic geometry and proves many facts which can then serve as a basic stock for a deeper study of these subjects.