Fundamentals of Diophantine Geometry
Title | Fundamentals of Diophantine Geometry PDF eBook |
Author | S. Lang |
Publisher | Springer Science & Business Media |
Pages | 383 |
Release | 2013-06-29 |
Genre | Mathematics |
ISBN | 1475718101 |
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.
Fundamentals of Diophantine Geometry
Title | Fundamentals of Diophantine Geometry PDF eBook |
Author | Serge Lang |
Publisher | |
Pages | 370 |
Release | 1983-01-01 |
Genre | Arithmetical algebraic geometry |
ISBN | 9783540908371 |
Number Theory III
Title | Number Theory III PDF eBook |
Author | Serge Lang |
Publisher | Springer Science & Business Media |
Pages | 68 |
Release | 1997-04-14 |
Genre | Mathematics |
ISBN | 9783540612230 |
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.
Diophantine Geometry
Title | Diophantine Geometry PDF eBook |
Author | Serge Lang |
Publisher | |
Pages | 192 |
Release | 1962 |
Genre | Diophantine analysis |
ISBN |
Fundamentals of Differential Geometry
Title | Fundamentals of Differential Geometry PDF eBook |
Author | Serge Lang |
Publisher | Springer Science & Business Media |
Pages | 553 |
Release | 2012-12-06 |
Genre | Mathematics |
ISBN | 1461205417 |
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
Diophantine Geometry
Title | Diophantine Geometry PDF eBook |
Author | Umberto Zannier |
Publisher | Springer |
Pages | 420 |
Release | 2007-06-27 |
Genre | Mathematics |
ISBN |
This book contains research articles on Diophantine Geometry, written by participants of a research program held at the Ennio De Giorgi Mathematical Research Center in Pisa, Italy, between April and July 2005. The authors are eminent experts in the field and present several subfields of the main topic. The volume provides a broad overview of recent research developments.
Diophantine Geometry
Title | Diophantine Geometry PDF eBook |
Author | Marc Hindry |
Publisher | Springer Science & Business Media |
Pages | 574 |
Release | 2013-12-01 |
Genre | Mathematics |
ISBN | 1461212103 |
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.