Logarithmic Forms and Diophantine Geometry

Logarithmic Forms and Diophantine Geometry
Author :
Publisher : Cambridge University Press
Total Pages :
Release :
ISBN-10 : 9781139468879
ISBN-13 : 1139468871
Rating : 4/5 (79 Downloads)

Synopsis Logarithmic Forms and Diophantine Geometry by : A. Baker

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.

Elliptic Diophantine Equations

Elliptic Diophantine Equations
Author :
Publisher : Walter de Gruyter
Total Pages : 196
Release :
ISBN-10 : 9783110281149
ISBN-13 : 3110281147
Rating : 4/5 (49 Downloads)

Synopsis Elliptic Diophantine Equations by : Nikos Tzanakis

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.

Heights in Diophantine Geometry

Heights in Diophantine Geometry
Author :
Publisher : Cambridge University Press
Total Pages : 676
Release :
ISBN-10 : 0521712297
ISBN-13 : 9780521712293
Rating : 4/5 (97 Downloads)

Synopsis Heights in Diophantine Geometry by : Enrico Bombieri

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

Diophantine Approximation on Linear Algebraic Groups

Diophantine Approximation on Linear Algebraic Groups
Author :
Publisher : Springer Science & Business Media
Total Pages : 649
Release :
ISBN-10 : 9783662115695
ISBN-13 : 3662115697
Rating : 4/5 (95 Downloads)

Synopsis Diophantine Approximation on Linear Algebraic Groups by : Michel Waldschmidt

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.

A Panorama of Number Theory Or The View from Baker's Garden

A Panorama of Number Theory Or The View from Baker's Garden
Author :
Publisher : Cambridge University Press
Total Pages : 378
Release :
ISBN-10 : 0521807999
ISBN-13 : 9780521807999
Rating : 4/5 (99 Downloads)

Synopsis A Panorama of Number Theory Or The View from Baker's Garden by : Gisbert Wüstholz

This is a selection of high quality articles on number theory by leading figures.

Arithmetic Geometry, Number Theory, and Computation

Arithmetic Geometry, Number Theory, and Computation
Author :
Publisher : Springer Nature
Total Pages : 587
Release :
ISBN-10 : 9783030809140
ISBN-13 : 3030809145
Rating : 4/5 (40 Downloads)

Synopsis Arithmetic Geometry, Number Theory, and Computation by : Jennifer S. Balakrishnan

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.

Transcendental Number Theory

Transcendental Number Theory
Author :
Publisher : Cambridge University Press
Total Pages : 180
Release :
ISBN-10 : 052139791X
ISBN-13 : 9780521397919
Rating : 4/5 (1X Downloads)

Synopsis Transcendental Number Theory by : Alan Baker

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.

An Invitation to Arithmetic Geometry

An Invitation to Arithmetic Geometry
Author :
Publisher : American Mathematical Society
Total Pages : 397
Release :
ISBN-10 : 9781470467258
ISBN-13 : 1470467259
Rating : 4/5 (58 Downloads)

Synopsis An Invitation to Arithmetic Geometry by : Dino Lorenzini

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.

Exponential Diophantine Equations

Exponential Diophantine Equations
Author :
Publisher : Cambridge University Press
Total Pages : 0
Release :
ISBN-10 : 0521091705
ISBN-13 : 9780521091701
Rating : 4/5 (05 Downloads)

Synopsis Exponential Diophantine Equations by : T. N. Shorey

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.

Lectures on Arakelov Geometry

Lectures on Arakelov Geometry
Author :
Publisher : Cambridge University Press
Total Pages : 190
Release :
ISBN-10 : 0521477093
ISBN-13 : 9780521477093
Rating : 4/5 (93 Downloads)

Synopsis Lectures on Arakelov Geometry by : C. Soulé

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