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| Author | : Enrico Bombieri |
| Publisher | : Cambridge University Press |
| Total Pages | : 676 |
| Release | : 2006 |
| Genre | : Mathematics |
| ISBN | : 9780521712293 |
This monograph is a bridge between the classical theory and modern approach via arithmetic geometry.
| Author | : Marc Hindry |
| Publisher | : Springer Science & Business Media |
| Total 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.
| Author | : S. Lang |
| Publisher | : Springer Science & Business Media |
| Total 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.
| Author | : Hideaki Ikoma |
| Publisher | : Cambridge University Press |
| Total Pages | : 179 |
| Release | : 2022-02-03 |
| Genre | : Mathematics |
| ISBN | : 1108845959 |
This book provides a self-contained proof of the Mordell conjecture (Faltings's theorem) and a concise introduction to Diophantine geometry.
| Author | : Emmanuel Peyre |
| Publisher | : Springer Nature |
| Total Pages | : 469 |
| Release | : 2021-03-10 |
| Genre | : Mathematics |
| ISBN | : 3030575594 |
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 | : A. Baker |
| Publisher | : Cambridge University Press |
| Total Pages | : |
| Release | : 2008-01-17 |
| Genre | : Mathematics |
| ISBN | : 1139468871 |
There is now much interplay between studies on logarithmic forms and deep aspects of arithmetic algebraic geometry. New light has been shed, for instance, on the famous conjectures of Tate and Shafarevich relating to abelian varieties and the associated celebrated discoveries of Faltings establishing the Mordell conjecture. This book gives an account of the theory of linear forms in the logarithms of algebraic numbers with special emphasis on the important developments of the past twenty-five years. The first part covers basic material in transcendental number theory but with a modern perspective. The remainder assumes some background in Lie algebras and group varieties, and covers, in some instances for the first time in book form, several advanced topics. The final chapter summarises other aspects of Diophantine geometry including hypergeometric theory and the André-Oort conjecture. A comprehensive bibliography rounds off this definitive survey of effective methods in Diophantine geometry.
| Author | : R. C. Mason |
| Publisher | : Cambridge University Press |
| Total Pages | : 142 |
| Release | : 1984-04-26 |
| Genre | : Mathematics |
| ISBN | : 9780521269834 |
A self-contained account of a new approach to the subject.
| Author | : Wolfgang M. Schmidt |
| Publisher | : Springer |
| Total Pages | : 224 |
| Release | : 2006-12-08 |
| Genre | : Mathematics |
| ISBN | : 3540473742 |
"This book by a leading researcher and masterly expositor of the subject studies diophantine approximations to algebraic numbers and their applications to diophantine equations. The methods are classical, and the results stressed can be obtained without much background in algebraic geometry. In particular, Thue equations, norm form equations and S-unit equations, with emphasis on recent explicit bounds on the number of solutions, are included. The book will be useful for graduate students and researchers." (L'Enseignement Mathematique) "The rich Bibliography includes more than hundred references. The book is easy to read, it may be a useful piece of reading not only for experts but for students as well." Acta Scientiarum Mathematicarum
| Author | : Umberto Zannier |
| Publisher | : Princeton University Press |
| Total Pages | : 175 |
| Release | : 2012-03-25 |
| Genre | : Mathematics |
| ISBN | : 1400842719 |
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).
| Author | : Nikos Tzanakis |
| Publisher | : Walter de Gruyter |
| Total Pages | : 196 |
| Release | : 2013-08-29 |
| Genre | : Mathematics |
| ISBN | : 3110281147 |
This book presents in a unified and concrete way the beautiful and deep mathematics - both theoretical and computational - on which the explicit solution of an elliptic Diophantine equation is based. It collects numerous results and methods that are scattered in the literature. Some results are hidden behind a number of routines in software packages, like Magma and Maple; professional mathematicians very often use these routines just as a black-box, having little idea about the mathematical treasure behind them. Almost 20 years have passed since the first publications on the explicit solution of elliptic Diophantine equations with the use of elliptic logarithms. The "art" of solving this type of equation has now reached its full maturity. The author is one of the main persons that contributed to the development of this art. The monograph presents a well-balanced combination of a variety of theoretical tools (from Diophantine geometry, algebraic number theory, theory of linear forms in logarithms of various forms - real/complex and p-adic elliptic - and classical complex analysis), clever computational methods and techniques (LLL algorithm and de Weger's reduction technique, AGM algorithm, Zagier's technique for computing elliptic integrals), ready-to-use computer packages. A result is the solution in practice of a large general class of Diophantine equations.
