Rational Points

Author : Joseph H. Silverman
Publisher : Springer Science & Business Media
ISBN 13 : 1475742525
Total Pages : 292 pages
Book Rating : 4.27/5 ( download)

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Download or read book Rational Points on Elliptic Curves written by Joseph H. Silverman and published by Springer Science & Business Media. This book was released on 2013-04-17 with total page 292 pages. Available in PDF, EPUB and Kindle. Book excerpt: The theory of elliptic curves involves a blend of algebra, geometry, analysis, and number theory. This book stresses this interplay as it develops the basic theory, providing an opportunity for readers to appreciate the unity of modern mathematics. The book’s accessibility, the informal writing style, and a wealth of exercises make it an ideal introduction for those interested in learning about Diophantine equations and arithmetic geometry.

Author : Bjorn Poonen
Publisher : American Mathematical Society
ISBN 13 : 1470474581
Total Pages : 357 pages
Book Rating : 4.84/5 ( download)

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Download or read book Rational Points on Varieties written by Bjorn Poonen and published by American Mathematical Society. This book was released on 2023-08-10 with total page 357 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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. The origins of arithmetic (or Diophantine) geometry can be traced back to antiquity, and it remains a lively and wide research domain up to our days. The book by Bjorn Poonen, a leading expert in the field, opens doors to this vast field for many readers with different experiences and backgrounds. It leads through various algebraic geometric constructions towards its central subject: obstructions to existence of rational points. —Yuri Manin, Max-Planck-Institute, Bonn It is clear that my mathematical life would have been very different if a book like this had been around at the time I was a student. —Hendrik Lenstra, University Leiden Understanding rational points on arbitrary algebraic varieties is the ultimate challenge. We have conjectures but few results. Poonen's book, with its mixture of basic constructions and openings into current research, will attract new generations to the Queen of Mathematics. —Jean-Louis Colliot-Thélène, Université Paris-Sud A beautiful subject, handled by a master. —Joseph Silverman, Brown University

Author : Jakob Stix
Publisher : Springer
ISBN 13 : 3642306748
Total Pages : 257 pages
Book Rating : 4.47/5 ( download)

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Download or read book Rational Points and Arithmetic of Fundamental Groups written by Jakob Stix and published by Springer. This book was released on 2012-10-19 with total page 257 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Author : Henri Darmon
Publisher : American Mathematical Soc.
ISBN 13 : 0821828681
Total Pages : 146 pages
Book Rating : 4.87/5 ( download)

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Download or read book Rational Points on Modular Elliptic Curves written by Henri Darmon and published by American Mathematical Soc.. This book was released on 2004 with total page 146 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book surveys some recent developments in the arithmetic of modular elliptic curves. It places a special emphasis on the construction of rational points on elliptic curves, the Birch and Swinnerton-Dyer conjecture, and the crucial role played by modularity in shedding light on these two closely related issues. The main theme of the book is the theory of complex multiplication, Heegner points, and some conjectural variants. The first three chapters introduce the background and prerequisites: elliptic curves, modular forms and the Shimura-Taniyama-Weil conjecture, complex multiplication and the Heegner point construction. The next three chapters introduce variants of modular parametrizations in which modular curves are replaced by Shimura curves attached to certain indefinite quaternion algebras. The main new contributions are found in Chapters 7-9, which survey the author's attempts to extend the theory of Heegner points and complex multiplication to situations where the base field is not a CM field. Chapter 10 explains the proof of Kolyvagin's theorem, which relates Heegner points to the arithmetic of elliptic curves and leads to the best evidence so far for the Birch and Swinnerton-Dyer conjecture.

Author : Alexei Skorobogatov
Publisher : Cambridge University Press
ISBN 13 : 0521802377
Total Pages : 197 pages
Book Rating : 4.76/5 ( download)

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Download or read book Torsors and Rational Points written by Alexei Skorobogatov and published by Cambridge University Press. This book was released on 2001-07-05 with total page 197 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book, first published in 2001, is a complete and coherent exposition of the theory and applications of torsors to rational points.

Author : Emmanuel Peyre
Publisher : Birkhäuser
ISBN 13 : 3034883684
Total Pages : 455 pages
Book Rating : 4.89/5 ( download)

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Download or read book Rational Points on Algebraic Varieties written by Emmanuel Peyre and published by Birkhäuser. This book was released on 2012-12-06 with total page 455 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Author : Gerd Faltings
Publisher : Springer Science & Business Media
ISBN 13 : 3663068129
Total Pages : 276 pages
Book Rating : 4.29/5 ( download)

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Download or read book Rational Points written by Gerd Faltings and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 276 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Author : Kunihiko Kodaira
Publisher : CUP Archive
ISBN 13 : 9780521217774
Total Pages : 424 pages
Book Rating : 4.76/5 ( download)

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Download or read book Complex Analysis and Algebraic Geometry written by Kunihiko Kodaira and published by CUP Archive. This book was released on 1977 with total page 424 pages. Available in PDF, EPUB and Kindle. Book excerpt: The articles in this volume cover some developments in complex analysis and algebraic geometry. The book is divided into three parts. Part I includes topics in the theory of algebraic surfaces and analytic surface. Part II covers topics in moduli and classification problems, as well as structure theory of certain complex manifolds. Part III is devoted to various topics in algebraic geometry analysis and arithmetic. A survey article by Ueno serves as an introduction to the general background of the subject matter of the volume. The volume was written for Kunihiko Kodaira on the occasion of his sixtieth birthday, by his friends and students. Professor Kodaira was one of the world's leading mathematicians in algebraic geometry and complex manifold theory: and the contributions reflect those concerns.

Author : Windy Dryden
Publisher : Routledge
ISBN 13 : 1135449139
Total Pages : 207 pages
Book Rating : 4.31/5 ( download)

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Download or read book Rational Emotive Behaviour Therapy written by Windy Dryden and published by Routledge. This book was released on 2007-01-24 with total page 207 pages. Available in PDF, EPUB and Kindle. Book excerpt: Rational Emotive Behaviour Therapy (REBT) is practised all over the world and has many therapeutic, occupational and educational applications. Rational Emotive Behaviour Therapy: 100 Key Points and Techniques presents 100 main features of this system, to help therapists improve their practice. These essential points have been derived from the authors' own practice, and also from their experience as trainers and supervisors of novice rational emotive behaviour therapists. Beginning with an introduction outlining the basics of the approach, this book offers thorough coverage of all the vital topics, including: - therapeutic alliance issues - educational issues - dealing with clients' misconceptions about REBT - encouraging clients to work at change - dealing with obstacles to client change - using the system in a creative way. This concise and highly practical book will be invaluable to psychotherapists and counsellors in training and practice, ensuring comprehensive understanding of the REBT approach.

Author : Marc Hindry
Publisher : Springer Science & Business Media
ISBN 13 : 1461212103
Total Pages : 574 pages
Book Rating : 4.02/5 ( download)

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Download or read book Diophantine Geometry written by Marc Hindry and published by Springer Science & Business Media. This book was released on 2013-12-01 with total page 574 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is an introduction to diophantine geometry at the advanced graduate level. The book contains a proof of the Mordell conjecture which will make it quite attractive to graduate students and professional mathematicians. In each part of the book, the reader will find numerous exercises.