Logarithmic Forms And Diophantine Geometry
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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 | : A. Baker |
Publisher | : Cambridge University Press |
Total Pages | : 208 |
Release | : 2008-01-17 |
Genre | : Mathematics |
ISBN | : 9780521882682 |
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 | : 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 | : Michel Waldschmidt |
Publisher | : Springer Science & Business Media |
Total Pages | : 649 |
Release | : 2013-03-14 |
Genre | : Mathematics |
ISBN | : 3662115697 |
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 | : 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.
Author | : Gisbert Wüstholz |
Publisher | : Cambridge University Press |
Total Pages | : 378 |
Release | : 2002-09-26 |
Genre | : Mathematics |
ISBN | : 9780521807999 |
This is a selection of high quality articles on number theory by leading figures.
Author | : Jennifer S. Balakrishnan |
Publisher | : Springer Nature |
Total Pages | : 587 |
Release | : 2022-03-15 |
Genre | : Mathematics |
ISBN | : 3030809145 |
This volume contains articles related to the work of the Simons Collaboration “Arithmetic Geometry, Number Theory, and Computation.” The papers present mathematical results and algorithms necessary for the development of large-scale databases like the L-functions and Modular Forms Database (LMFDB). The authors aim to develop systematic tools for analyzing Diophantine properties of curves, surfaces, and abelian varieties over number fields and finite fields. The articles also explore examples important for future research. Specific topics include● algebraic varieties over finite fields● the Chabauty-Coleman method● modular forms● rational points on curves of small genus● S-unit equations and integral points.
Author | : Alan Baker |
Publisher | : Cambridge University Press |
Total Pages | : 185 |
Release | : 2022-06-09 |
Genre | : Computers |
ISBN | : 100922994X |
Alan Baker's systematic account of transcendental number theory, with a new introduction and afterword explaining recent developments.
Author | : Dino Lorenzini |
Publisher | : American Mathematical Society |
Total Pages | : 397 |
Release | : 2021-12-23 |
Genre | : Mathematics |
ISBN | : 1470467259 |
Extremely carefully written, masterfully thought out, and skillfully arranged introduction … to the arithmetic of algebraic curves, on the one hand, and to the algebro-geometric aspects of number theory, on the other hand. … an excellent guide for beginners in arithmetic geometry, just as an interesting reference and methodical inspiration for teachers of the subject … a highly welcome addition to the existing literature. —Zentralblatt MATH The interaction between number theory and algebraic geometry has been especially fruitful. In this volume, the author gives a unified presentation of some of the basic tools and concepts in number theory, commutative algebra, and algebraic geometry, and for the first time in a book at this level, brings out the deep analogies between them. The geometric viewpoint is stressed throughout the book. Extensive examples are given to illustrate each new concept, and many interesting exercises are given at the end of each chapter. Most of the important results in the one-dimensional case are proved, including Bombieri's proof of the Riemann Hypothesis for curves over a finite field. While the book is not intended to be an introduction to schemes, the author indicates how many of the geometric notions introduced in the book relate to schemes, which will aid the reader who goes to the next level of this rich subject.
Author | : Jan-Hendrik Evertse |
Publisher | : Cambridge University Press |
Total Pages | : 381 |
Release | : 2015-12-30 |
Genre | : Mathematics |
ISBN | : 1316432351 |
Diophantine number theory is an active area that has seen tremendous growth over the past century, and in this theory unit equations play a central role. This comprehensive treatment is the first volume devoted to these equations. The authors gather together all the most important results and look at many different aspects, including effective results on unit equations over number fields, estimates on the number of solutions, analogues for function fields and effective results for unit equations over finitely generated domains. They also present a variety of applications. Introductory chapters provide the necessary background in algebraic number theory and function field theory, as well as an account of the required tools from Diophantine approximation and transcendence theory. This makes the book suitable for young researchers as well as experts who are looking for an up-to-date overview of the field.