Fundamentals Of Diophantine Geometry

Author : S. Lang
Publisher : Springer Science & Business Media
Page : 383 pages
File Size : 42,7 Mb
Release : 2013-06-29
Category : Mathematics
ISBN : 9781475718102

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Fundamentals of Diophantine Geometry by S. Lang Pdf

Diophantine problems represent some of the strongest aesthetic attractions to algebraic geometry. They consist in giving criteria for the existence of solutions of algebraic equations in rings and fields, and eventually for the number of such solutions. The fundamental ring of interest is the ring of ordinary integers Z, and the fundamental field of interest is the field Q of rational numbers. One discovers rapidly that to have all the technical freedom needed in handling general problems, one must consider rings and fields of finite type over the integers and rationals. Furthermore, one is led to consider also finite fields, p-adic fields (including the real and complex numbers) as representing a localization of the problems under consideration. We shall deal with global problems, all of which will be of a qualitative nature. On the one hand we have curves defined over say the rational numbers. Ifthe curve is affine one may ask for its points in Z, and thanks to Siegel, one can classify all curves which have infinitely many integral points. This problem is treated in Chapter VII. One may ask also for those which have infinitely many rational points, and for this, there is only Mordell's conjecture that if the genus is :;;; 2, then there is only a finite number of rational points.

Author : Serge Lang
Publisher : Unknown
Page : 370 pages
File Size : 45,5 Mb
Release : 1983-01-01
Category : Arithmetical algebraic geometry
ISBN : 3540908374

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Fundamentals of Diophantine Geometry by Serge Lang Pdf

Author : Serge Lang
Publisher : Unknown
Page : 192 pages
File Size : 53,9 Mb
Release : 1962
Category : Diophantine analysis
ISBN : STANFORD:36105031176063

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Diophantine Geometry by Serge Lang Pdf

Author : Serge Lang
Publisher : Springer Science & Business Media
Page : 68 pages
File Size : 49,8 Mb
Release : 1997-04-14
Category : Mathematics
ISBN : 3540612238

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Number Theory III by Serge Lang Pdf

In 1988 Shafarevich asked me to write a volume for the Encyclopaedia of Mathematical Sciences on Diophantine Geometry. I said yes, and here is the volume. By definition, diophantine problems concern the solutions of equations in integers, or rational numbers, or various generalizations, such as finitely generated rings over Z or finitely generated fields over Q. The word Geometry is tacked on to suggest geometric methods. This means that the present volume is not elementary. For a survey of some basic problems with a much more elementary approach, see [La 9Oc]. The field of diophantine geometry is now moving quite rapidly. Out standing conjectures ranging from decades back are being proved. I have tried to give the book some sort of coherence and permanence by em phasizing structural conjectures as much as results, so that one has a clear picture of the field. On the whole, I omit proofs, according to the boundary conditions of the encyclopedia. On some occasions I do give some ideas for the proofs when these are especially important. In any case, a lengthy bibliography refers to papers and books where proofs may be found. I have also followed Shafarevich's suggestion to give examples, and I have especially chosen these examples which show how some classical problems do or do not get solved by contemporary in sights. Fermat's last theorem occupies an intermediate position. Al though it is not proved, it is not an isolated problem any more.

Author : Serge Lang
Publisher : Springer Science & Business Media
Page : 553 pages
File Size : 45,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461205418

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Fundamentals of Differential Geometry by Serge Lang Pdf

This book provides an introduction to the basic concepts in differential topology, differential geometry, and differential equations, and some of the main basic theorems in all three areas. This new edition includes new chapters, sections, examples, and exercises. From the reviews: "There are many books on the fundamentals of differential geometry, but this one is quite exceptional; this is not surprising for those who know Serge Lang's books." --EMS NEWSLETTER

Author : Serge Lang
Publisher : Hassell Street Press
Page : 0 pages
File Size : 50,9 Mb
Release : 2021-09-09
Category : Mathematics
ISBN : 1013889118

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Diophantine Geometry by Serge Lang Pdf

This work has been selected by scholars as being culturally important and is part of the knowledge base of civilization as we know it. This work is in the public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. To ensure a quality reading experience, this work has been proofread and republished using a format that seamlessly blends the original graphical elements with text in an easy-to-read typeface. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant.

Author : Marc Hindry,Joseph H. Silverman
Publisher : Springer Science & Business Media
Page : 574 pages
File Size : 40,8 Mb
Release : 2013-12-01
Category : Mathematics
ISBN : 9781461212102

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Diophantine Geometry by Marc Hindry,Joseph H. Silverman Pdf

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.

Author : N.B. Singh
Publisher : N.B. Singh
Page : 63 pages
File Size : 46,5 Mb
Release : 2024-06-30
Category : Mathematics
ISBN : 8210379456XXX

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Diophantine Discoveries Fundamentals by N.B. Singh Pdf

"Diophantine Discoveries Fundamentals" is a beginner-friendly exploration of the captivating world of Diophantine equations, designed for those with no prior mathematical background. Delving into the realm of mathematical puzzles, this book offers clear and accessible explanations of Diophantine equations, starting from the basics and gradually building up the reader's understanding. Through engaging examples and straightforward language, readers are introduced to the fascinating concepts of finding whole number solutions to polynomial equations. From the historical significance of Diophantine equations to their applications in various fields such as number theory, algebra, and cryptography, this book serves as an inviting gateway for curious minds to unravel the mysteries of mathematics. Whether you're a student eager to expand your mathematical knowledge or simply someone with a passion for learning, "Diophantine Discoveries Fundamentals" provides an enjoyable and educational journey into the heart of mathematical exploration.

Author : Enrico Bombieri,Walter Gubler
Publisher : Cambridge University Press
Page : 73 pages
File Size : 51,5 Mb
Release : 2007-09-06
Category : Mathematics
ISBN : 9781139447959

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Heights in Diophantine Geometry by Enrico Bombieri,Walter Gubler Pdf

Diophantine geometry has been studied by number theorists for thousands of years, this monograph is a bridge between the classical theory and modern approach via arithmetic geometry. The treatment is largely self-contained, with proofs given in full detail. Many results appear here for the first time.

Author : Serge Lang
Publisher : Springer Science & Business Media
Page : 492 pages
File Size : 46,5 Mb
Release : 2000-07-28
Category : Mathematics
ISBN : 0387988041

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Collected Papers IV by Serge Lang Pdf

Serge Lang is not only one of the top mathematicians of our time, but also an excellent writer. He has made innumerable and invaluable contributions in diverse fields of mathematics and was honoured with the Cole Prize by the American Mathematical Society as well as with the Prix Carriere by the French Academy of Sciences. Here, 83 of his research papers are collected in four volumes, ranging over a variety of topics of interest to many readers.

Author : Pietro Corvaja
Publisher : Unknown
Page : 0 pages
File Size : 45,7 Mb
Release : 2016
Category : Arithmetical algebraic geometry
ISBN : 9380250835

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Integral Points on Algebraic Varieties by Pietro Corvaja Pdf

Offers an introduction to Diophantine Geometry. The central theme is the investigation of the distribution of integral points on algebraic varieties. The text 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.

Author : Jorn Steuding
Publisher : CRC Press
Page : 275 pages
File Size : 48,6 Mb
Release : 2005-05-19
Category : Mathematics
ISBN : 9781584884828

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Diophantine Analysis by Jorn Steuding Pdf

While its roots reach back to the third century, diophantine analysis continues to be an extremely active and powerful area of number theory. Many diophantine problems have simple formulations, they can be extremely difficult to attack, and many open problems and conjectures remain. Diophantine Analysis examines the theory of diophantine approximations and the theory of diophantine equations, with emphasis on interactions between these subjects. Beginning with the basic principles, the author develops his treatment around the theory of continued fractions and examines the classic theory, including some of its applications. He also explores modern topics rarely addressed in other texts, including the abc conjecture, the polynomial Pell equation, and the irrationality of the zeta function and touches on topics and applications related to discrete mathematics, such as factoring methods for large integers. Setting the stage for tackling the field's many open problems and conjectures, Diophantine Analysis is an ideal introduction to the fundamentals of this venerable but still dynamic field. A detailed appendix supplies the necessary background material, more than 200 exercises reinforce the concepts, and engaging historical notes bring the subject to life.

Author : Takushiro Ochiai,Toshiki Mabuchi,Yoshiaki Maeda,Junjiro Noguchi,Alan Weinstein
Publisher : Springer
Page : 481 pages
File Size : 50,6 Mb
Release : 2015-02-25
Category : Mathematics
ISBN : 9783319115238

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Geometry and Analysis on Manifolds by Takushiro Ochiai,Toshiki Mabuchi,Yoshiaki Maeda,Junjiro Noguchi,Alan Weinstein Pdf

This volume is dedicated to the memory of Shoshichi Kobayashi, and gathers contributions from distinguished researchers working on topics close to his research areas. The book is organized into three parts, with the first part presenting an overview of Professor Shoshichi Kobayashi’s career. This is followed by two expository course lectures (the second part) on recent topics in extremal Kähler metrics and value distribution theory, which will be helpful for graduate students in mathematics interested in new topics in complex geometry and complex analysis. Lastly, the third part of the volume collects authoritative research papers on differential geometry and complex analysis. Professor Shoshichi Kobayashi was a recognized international leader in the areas of differential and complex geometry. He contributed crucial ideas that are still considered fundamental in these fields. The book will be of interest to researchers in the fields of differential geometry, complex geometry, and several complex variables geometry, as well as to graduate students in mathematics.

Author : Michel Waldschmidt
Publisher : Springer Science & Business Media
Page : 649 pages
File Size : 46,5 Mb
Release : 2013-03-14
Category : Mathematics
ISBN : 9783662115695

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Diophantine Approximation on Linear Algebraic Groups by Michel Waldschmidt Pdf

The theory of transcendental numbers is closely related to the study of diophantine approximation. This book deals with values of the usual exponential function ez: a central open problem is the conjecture on algebraic independence of logarithms of algebraic numbers. Two chapters provide complete and simplified proofs of zero estimates (due to Philippon) on linear algebraic groups.

Author : Umberto Zannier
Publisher : Springer
Page : 169 pages
File Size : 47,6 Mb
Release : 2015-02-13
Category : Mathematics
ISBN : 9788876425202

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On Some Applications of Diophantine Approximations by Umberto Zannier Pdf

This book consists mainly of the translation, by C. Fuchs, of the 1929 landmark paper "Über einige Anwendungen diophantischer Approximationen" by C.L. Siegel. The paper contains proofs of most important results in transcendence theory and diophantine analysis, notably Siegel’s celebrated theorem on integral points on algebraic curves. Many modern versions of Siegel’s proof have appeared, but none seem to faithfully reproduce all features of the original one. This translation makes Siegel’s original ideas and proofs available for the first time in English. The volume also contains the original version of the paper (in German) and an article by the translator and U. Zannier, commenting on some aspects of the evolution of this field following Siegel’s paper. To end, it presents three modern proofs of Siegel’s theorem on integral points.