**Author**: Enrico Bombieri

**Publisher:** Cambridge University Press

**ISBN:**

**Category:** Mathematics

**Page:** 652

**View:** 366

- 2007-09-06
- in Mathematics
- Enrico Bombieri

**Author**: Enrico Bombieri

**Publisher:** Cambridge University Press

**ISBN:**

**Category:** Mathematics

**Page:** 652

**View:** 366

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

- 2010-07-23
- in
- Enrico Bombieri

**Author**: Enrico Bombieri

**Publisher:**

**ISBN:**

**Category:**

**Page:**

**View:** 863

- 2006
- in Arithmetical algebraic geometry
- Enrico Bombieri

**Author**: Enrico Bombieri

**Publisher:**

**ISBN:**

**Category:** Arithmetical algebraic geometry

**Page:** 652

**View:** 684

Diophantine geometry has been studied by number theorists for thousands of years, this monograph is a bridge between the classical theory and modern approach via arithmetic geometry. The treatment is largely self-contained, with proofs given in full detail. Many results appear here for the first time.

- 2013-12-01
- in Mathematics
- Marc Hindry

*An Introduction*

**Author**: Marc Hindry

**Publisher:** Springer Science & Business Media

**ISBN:**

**Category:** Mathematics

**Page:** 561

**View:** 253

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.

- 2013-06-29
- in Mathematics
- S. Lang

**Author**: S. Lang

**Publisher:** Springer Science & Business Media

**ISBN:**

**Category:** Mathematics

**Page:** 370

**View:** 735

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.

- 2016-03-02
- in Mathematics
- Lars Halvard Halle

**Author**: Lars Halvard Halle

**Publisher:** Springer

**ISBN:**

**Category:** Mathematics

**Page:** 151

**View:** 284

Presenting the first systematic treatment of the behavior of Néron models under ramified base change, this book can be read as an introduction to various subtle invariants and constructions related to Néron models of semi-abelian varieties, motivated by concrete research problems and complemented with explicit examples. Néron models of abelian and semi-abelian varieties have become an indispensable tool in algebraic and arithmetic geometry since Néron introduced them in his seminal 1964 paper. Applications range from the theory of heights in Diophantine geometry to Hodge theory. We focus specifically on Néron component groups, Edixhoven’s filtration and the base change conductor of Chai and Yu, and we study these invariants using various techniques such as models of curves, sheaves on Grothendieck sites and non-archimedean uniformization. We then apply our results to the study of motivic zeta functions of abelian varieties. The final chapter contains a list of challenging open questions. This book is aimed towards researchers with a background in algebraic and arithmetic geometry.

- 2022
- in MATHEMATICS
- Atsushi Moriwaki

*A Complete Proof from Diophantine Geometry*

**Author**: Atsushi Moriwaki

**Publisher:**

**ISBN:**

**Category:** MATHEMATICS

**Page:**

**View:** 317

"The Mordell conjecture (Faltings's theorem) is one of the most important achievements in Diophantine geometry, stating that an algebraic curve of genus at least two has only finitely many rational points. This book provides a self-contained and detailed proof of the Mordell conjecture following the papers of Bombieri and Vojta. Also acting as a concise introduction to Diophantine geometry, the text starts from basics of algebraic number theory, touches on several important theorems and techniques (including the theory of heights, the Mordell- Weil theorem, Siegel's lemma and Roth's lemma) from Diophantine geometry, and culminates in the proof of the Mordell conjecture. Based on the authors' own teaching experience, it will be of great value to advanced undergraduate and graduate students in algebraic geometry and number theory, as well as researchers interested in Diophantine geometry as a whole"--

- 2007-06-27
- in Mathematics
- Umberto Zannier

*Proceedings*

**Author**: Umberto Zannier

**Publisher:** Springer

**ISBN:**

**Category:** Mathematics

**Page:** 390

**View:** 600

This book contains research articles on Diophantine Geometry, written by participants of a research program held at the Ennio De Giorgi Mathematical Research Center in Pisa, Italy, between April and July 2005. The authors are eminent experts in the field and present several subfields of the main topic. The volume provides a broad overview of recent research developments.

- 2021
- in Arakelov theory
- Emmanuel Peyre

**Author**: Emmanuel Peyre

**Publisher:** Springer Nature

**ISBN:**

**Category:** Arakelov theory

**Page:** 469

**View:** 693

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.

- 2013-12-01
- in Mathematics
- Michiel Hazewinkel

*Volume 6: Subject Index — Author Index*

**Author**: Michiel Hazewinkel

**Publisher:** Springer Science & Business Media

**ISBN:**

**Category:** Mathematics

**Page:** 732

**View:** 625

This ENCYCLOPAEDIA OF MATHEMATICS aims to be a reference work for all parts of mathe matics. It is a translation with updates and editorial comments of the Soviet Mathematical Encyclopaedia published by 'Soviet Encyclopaedia Publishing House' in five volumes in 1977-1985. The annotated translation consists of ten volumes including a special index volume. There are three kinds of articles in this ENCYCLOPAEDIA. First of all there are survey-type articles dealing with the various main directions in mathematics (where a rather fine subdivi sion has been used). The main requirement for these articles has been that they should give a reasonably complete up-to-date account of the current state of affairs in these areas and that they should be maximally accessible. On the whole, these articles should be understandable to mathematics students in their first specialization years, to graduates from other mathematical areas and, depending on the specific subject, to specialists in other domains of science, en gineers and teachers of mathematics. These articles treat their material at a fairly general level and aim to give an idea of the kind of problems, techniques and concepts involved in the area in question. They also contain background and motivation rather than precise statements of precise theorems with detailed definitions and technical details on how to carry out proofs and constructions. The second kind of article, of medium length, contains more detailed concrete problems, results and techniques.

- 2013-06-29
- in Technology & Engineering
- Jean-P. Serre

**Author**: Jean-P. Serre

**Publisher:** Springer Science & Business Media

**ISBN:**

**Category:** Technology & Engineering

**Page:** 218

**View:** 833

The book is based on a course given by J.-P. Serre at the Collège de France in 1980 and 1981. Basic techniques in Diophantine geometry are covered, such as heights, the Mordell-Weil theorem, Siegel's and Baker's theorems, Hilbert's irreducibility theorem, and the large sieve. Included are applications to, for example, Mordell's conjecture, the construction of Galois extensions, and the classical class number 1 problem. Comprehensive bibliographical references.

- 1962
- in Analyse diophantienne
- Serge Lang

**Author**: Serge Lang

**Publisher:** Hassell Street Press

**ISBN:**

**Category:** Analyse diophantienne

**Page:** 170

**View:** 192

This work has been selected by scholars as being culturally important and is part of the knowledge base of civilization as we know it. This work is in the public domain in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. To ensure a quality reading experience, this work has been proofread and republished using a format that seamlessly blends the original graphical elements with text in an easy-to-read typeface. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant.

- 2010-05-05
- in Mathematics
- J.H. Silverman

**Author**: J.H. Silverman

**Publisher:** Springer Science & Business Media

**ISBN:**

**Category:** Mathematics

**Page:** 511

**View:** 425

This book provides an introduction to the relatively new discipline of arithmetic dynamics. Whereas classical discrete dynamics is the study of iteration of self-maps of the complex plane or real line, arithmetic dynamics is the study of the number-theoretic properties of rational and algebraic points under repeated application of a polynomial or rational function. A principal theme of arithmetic dynamics is that many of the fundamental problems in the theory of Diophantine equations have dynamical analogs.This graduate-level text provides an entry for students into an active field of research and serves as a standard reference for researchers.

- 2017-05-26
- in Mathematics
- Christian Elsholtz

*Festschrift in Honour of Robert F. Tichy’s 60th Birthday*

**Author**: Christian Elsholtz

**Publisher:** Springer

**ISBN:**

**Category:** Mathematics

**Page:** 444

**View:** 768

This volume is dedicated to Robert F. Tichy on the occasion of his 60th birthday. Presenting 22 research and survey papers written by leading experts in their respective fields, it focuses on areas that align with Tichy’s research interests and which he significantly shaped, including Diophantine problems, asymptotic counting, uniform distribution and discrepancy of sequences (in theory and application), dynamical systems, prime numbers, and actuarial mathematics. Offering valuable insights into recent developments in these areas, the book will be of interest to researchers and graduate students engaged in number theory and its applications.

- 2015-08-13
- in Mathematics
- A. J. Wilkie

**Author**: A. J. Wilkie

**Publisher:** Cambridge University Press

**ISBN:**

**Category:** Mathematics

**Page:** 232

**View:** 618

Brings the researcher up to date with recent applications of mathematical logic to number theory.

- 2016-10-10
- in Mathematics
- Jan-Hendrik Evertse

**Author**: Jan-Hendrik Evertse

**Publisher:** Cambridge University Press

**ISBN:**

**Category:** Mathematics

**Page:** 454

**View:** 769

The first comprehensive and up-to-date account of discriminant equations and their applications. For graduate students and researchers.

- 2013-04-17
- in Mathematics
- Gerd Faltings

**Author**: Gerd Faltings

**Publisher:** Springer Science & Business Media

**ISBN:**

**Category:** Mathematics

**Page:** 318

**View:** 552

A new and complete treatment of semi-abelian degenerations of abelian varieties, and their application to the construction of arithmetic compactifications of Siegel moduli space, with most of the results being published for the first time. Highlights of the book include a classification of semi-abelian schemes, construction of the toroidal and the minimal compactification over the integers, heights for abelian varieties over number fields, and Eichler integrals in several variables, together with a new approach to Siegel modular forms. A valuable source of reference for researchers and graduate students interested in algebraic geometry, Shimura varieties or diophantine geometry.

- 2015-07-16
- in Computers
- L. Beshaj

**Author**: L. Beshaj

**Publisher:** IOS Press

**ISBN:**

**Category:** Computers

**Page:** 388

**View:** 789

This book had its origins in the NATO Advanced Study Institute (ASI) held in Ohrid, Macedonia, in 2014. The focus of this ASI was the arithmetic of superelliptic curves and their application in different scientific areas, including whether all the applications of hyperelliptic curves, such as cryptography, mathematical physics, quantum computation and diophantine geometry, can be carried over to the superelliptic curves. Additional papers have been added which provide some background for readers who were not at the conference, with the intention of making the book logically more complete and easier to read, but familiarity with the basic facts of algebraic geometry, commutative algebra and number theory are assumed. The book is divided into three sections. The first part deals with superelliptic curves with regard to complex numbers, the automorphisms group and the corresponding Hurwitz loci. The second part of the book focuses on the arithmetic of the subject, while the third addresses some of the applications of superelliptic curves.

- 2016-11-23
- in Mathematics
- Pietro Corvaja

*An Introduction to Diophantine Geometry*

**Author**: Pietro Corvaja

**Publisher:** Springer

**ISBN:**

**Category:** Mathematics

**Page:** 75

**View:** 610

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.

- 2014
- in
- Robert Vernon Lees Grizzard

**Author**: Robert Vernon Lees Grizzard

**Publisher:**

**ISBN:**

**Category:**

**Page:** 160

**View:** 317

This dissertation contains a number of results on properties of infinite algebraic extensions of the rational field, all of which have a view toward the study of heights in diophantine geometry. We investigate whether subextensions of extensions generated by roots of polynomials of a given degree are themselves generated by polynomials of small degree, a problem motivated by the study of heights. We discuss a relative version of the Bogomolov property (the absence of small points) for extensions of fields of algebraic numbers. We describe the relationship between the Bogomolov property and the structure of the multiplicative group. Finally, we describe some results on height lower bounds which can be interpreted as diophantine approximation results in the multiplicative group.