Heights in Diophantine Geometry

Heights 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.

Unit Equations in Diophantine Number Theory

Unit Equations in Diophantine Number Theory
Author: Jan-Hendrik Evertse
Publisher: Cambridge University Press
Total Pages: 381
Release: 2015-12-30
Genre: Mathematics
ISBN: 1107097606

A comprehensive, graduate-level treatment of unit equations and their various applications.

Class Groups of Number Fields and Related Topics

Class Groups of Number Fields and Related Topics
Author: Kalyan Chakraborty
Publisher: Springer Nature
Total Pages: 182
Release: 2020-01-17
Genre: Mathematics
ISBN: 981151514X

This book gathers original research papers and survey articles presented at the “International Conference on Class Groups of Number Fields and Related Topics,” held at Harish-Chandra Research Institute, Allahabad, India, on September 4–7, 2017. It discusses the fundamental research problems that arise in the study of class groups of number fields and introduces new techniques and tools to study these problems. Topics in this book include class groups and class numbers of number fields, units, the Kummer–Vandiver conjecture, class number one problem, Diophantine equations, Thue equations, continued fractions, Euclidean number fields, heights, rational torsion points on elliptic curves, cyclotomic numbers, Jacobi sums, and Dedekind zeta values. This book is a valuable resource for undergraduate and graduate students of mathematics as well as researchers interested in class groups of number fields and their connections to other branches of mathematics. New researchers to the field will also benefit immensely from the diverse problems discussed. All the contributing authors are leading academicians, scientists, researchers, and scholars.

Diophantine Approximation and Abelian Varieties

Diophantine Approximation and Abelian Varieties
Author: Bas Edixhoven
Publisher: Springer
Total Pages: 136
Release: 2009-02-05
Genre: Mathematics
ISBN: 3540482083

The 13 chapters of this book centre around the proof of Theorem 1 of Faltings' paper "Diophantine approximation on abelian varieties", Ann. Math.133 (1991) and together give an approach to the proof that is accessible to Ph.D-level students in number theory and algebraic geometry. Each chapter is based on an instructional lecture given by its author ata special conference for graduate students, on the topic of Faltings' paper.

Integral Points on Algebraic Varieties

Integral Points on Algebraic Varieties
Author: Pietro Corvaja
Publisher: Springer
Total Pages: 82
Release: 2016-11-23
Genre: Mathematics
ISBN: 9811026483

This book is intended to be an introduction to Diophantine geometry. The central theme of the book is to investigate the distribution of integral points on algebraic varieties. This text rapidly introduces problems in Diophantine geometry, especially those involving integral points, assuming a geometrical perspective. It presents recent results not available in textbooks and also new viewpoints on classical material. In some instances, proofs have been replaced by a detailed analysis of particular cases, referring to the quoted papers for complete proofs. A central role is played by Siegel’s finiteness theorem for integral points on curves. The book ends with the analysis of integral points on surfaces.

European Congress of Mathematics

European Congress of Mathematics
Author: Ari Laptev
Publisher: European Mathematical Society
Total Pages: 906
Release: 2005
Genre: Mathematics
ISBN: 9783037190098

The European Congress of Mathematics, held every four years, has established itself as a major international mathematical event. Following those in Paris, 1992, Budapest, 1996, and Barcelona, 2000, the Fourth European Congress of Mathematics took place in Stockholm, Sweden, June 27 to July 2, 2004, with 913 participants from 65 countries. Apart from seven plenary and thirty three invited lectures, there were six Science Lectures covering the most relevant aspects of mathematics in science and technology. Moreover, twelve projects of the EU Research Training Networks in Mathematics and Information Sciences, as well as Programmes from the European Science Foundation in Physical and Engineering Sciences, were presented. Ten EMS Prizes were awarded to young European mathematicians who have made a particular contribution to the progress of mathematics. Five of the prizewinners were independently chosen by the 4ECM Scientific Committee as plenary or invited speakers. The other five prizewinners gave their lectures in parallel sessions. Most of these contributions are now collected in this volume, providing a permanent record of so much that is best in mathematics today.

On Some Applications of Diophantine Approximations

On Some Applications of Diophantine Approximations
Author: Umberto Zannier
Publisher: Springer
Total Pages: 169
Release: 2015-02-13
Genre: Mathematics
ISBN: 8876425209

This book consists mainly of the translation, by C. Fuchs, of the 1929 landmark paper "Über einige Anwendungen diophantischer Approximationen" by C.L. Siegel. The paper contains proofs of most important results in transcendence theory and diophantine analysis, notably Siegel’s celebrated theorem on integral points on algebraic curves. Many modern versions of Siegel’s proof have appeared, but none seem to faithfully reproduce all features of the original one. This translation makes Siegel’s original ideas and proofs available for the first time in English. The volume also contains the original version of the paper (in German) and an article by the translator and U. Zannier, commenting on some aspects of the evolution of this field following Siegel’s paper. To end, it presents three modern proofs of Siegel’s theorem on integral points.

Pillars of Transcendental Number Theory

Pillars of Transcendental Number Theory
Author: Saradha Natarajan
Publisher: Springer Nature
Total Pages: 184
Release: 2020-05-02
Genre: Mathematics
ISBN: 9811541558

This book deals with the development of Diophantine problems starting with Thue's path breaking result and culminating in Roth's theorem with applications. It discusses classical results including Hermite–Lindemann–Weierstrass theorem, Gelfond–Schneider theorem, Schmidt’s subspace theorem and more. It also includes two theorems of Ramachandra which are not widely known and other interesting results derived on the values of Weierstrass elliptic function. Given the constantly growing number of applications of linear forms in logarithms, it is becoming increasingly important for any student wanting to work in this area to know the proofs of Baker’s original results. This book presents Baker’s original results in a format suitable for graduate students, with a focus on presenting the content in an accessible and simple manner. Each student-friendly chapter concludes with selected problems in the form of “Exercises” and interesting information presented as “Notes,” intended to spark readers’ curiosity.

Diophantine Approximation on Linear Algebraic Groups

Diophantine Approximation on Linear Algebraic Groups
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.