Logarithmic Forms and Diophantine Geometry
Logarithmic Forms and Diophantine Geometry

Author: A. Baker

Publisher: Cambridge University Press

Published: 2008-01-17

Total Pages:

ISBN-13: 1139468871

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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.

Heights in Diophantine Geometry
Heights in Diophantine Geometry

Author: Enrico Bombieri

Publisher: Cambridge University Press

Published: 2006

Total Pages: 676

ISBN-13: 9780521712293

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This monograph is a bridge between the classical theory and modern approach via arithmetic geometry.

Arithmetic Geometry, Number Theory, and Computation
Arithmetic Geometry, Number Theory, and Computation

Author: Jennifer S. Balakrishnan

Publisher: Springer Nature

Published: 2022-03-15

Total Pages: 587

ISBN-13: 3030809145

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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.

An Invitation to Arithmetic Geometry
An Invitation to Arithmetic Geometry

Author: Dino Lorenzini

Publisher: American Mathematical Society

Published: 2021-12-23

Total Pages: 397

ISBN-13: 1470467259

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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.

Transcendental Number Theory
Transcendental Number Theory

Author: Alan Baker

Publisher: Cambridge University Press

Published: 1990-09-28

Total Pages: 180

ISBN-13: 9780521397919

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First published in 1975, this classic book gives a systematic account of transcendental number theory, that is those numbers which cannot be expressed as the roots of algebraic equations having rational coefficients. Their study has developed into a fertile and extensive theory enriching many branches of pure mathematics. Expositions are presented of theories relating to linear forms in the logarithms of algebraic numbers, of Schmidt's generalisation of the Thue-Siegel-Roth theorem, of Shidlovsky's work on Siegel's |E|-functions and of Sprindzuk's solution to the Mahler conjecture. The volume was revised in 1979: however Professor Baker has taken this further opportunity to update the book including new advances in the theory and many new references.

Exponential Diophantine Equations
Exponential Diophantine Equations

Author: T. N. Shorey

Publisher: Cambridge University Press

Published: 2008-12-04

Total Pages: 0

ISBN-13: 9780521091701

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This is a integrated presentation of the theory of exponential diophantine equations. The authors present, in a clear and unified fashion, applications to exponential diophantine equations and linear recurrence sequences of the Gelfond-Baker theory of linear forms in logarithms of algebraic numbers. Topics covered include the Thue equations, the generalised hyperelliptic equation, and the Fermat and Catalan equations. The necessary preliminaries are given in the first three chapters. Each chapter ends with a section giving details of related results.

Diophantus and Diophantine Equations
Diophantus and Diophantine Equations

Author: Isabella Grigoryevna Bashmakova

Publisher: American Mathematical Soc.

Published: 2019-01-29

Total Pages: 106

ISBN-13: 1470450496

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This book tells the story of Diophantine analysis, a subject that, owing to its thematic proximity to algebraic geometry, became fashionable in the last half century and has remained so ever since. This new treatment of the methods of Diophantus--a person whose very existence has long been doubted by most historians of mathematics--will be accessible to readers who have taken some university mathematics. It includes the elementary facts of algebraic geometry indispensable for its understanding. The heart of the book is a fascinating account of the development of Diophantine methods during the.

Lectures on Arakelov Geometry
Lectures on Arakelov Geometry

Author: C. Soulé

Publisher: Cambridge University Press

Published: 1994-09-15

Total Pages: 190

ISBN-13: 9780521477093

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An account for graduate students of this new technique in diophantine geometry; includes account of higher dimensional theory.

Unit Equations in Diophantine Number Theory
Unit Equations in Diophantine Number Theory

Author: Jan-Hendrik Evertse

Publisher: Cambridge University Press

Published: 2015-12-30

Total Pages: 381

ISBN-13: 1316432351

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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.