Rational Points on Elliptic Curves
Rational Points on Elliptic Curves

Author: Joseph H. Silverman

Publisher: Springer Science & Business Media

Published: 2013-04-17

Total Pages: 292

ISBN-13: 1475742525

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The theory of elliptic curves involves a blend of algebra, geometry, analysis, and number theory. This book stresses this interplay as it develops the basic theory, providing an opportunity for readers to appreciate the unity of modern mathematics. The book’s accessibility, the informal writing style, and a wealth of exercises make it an ideal introduction for those interested in learning about Diophantine equations and arithmetic geometry.

Rational Points on Varieties
Rational Points on Varieties

Author: Bjorn Poonen

Publisher: American Mathematical Soc.

Published: 2017-12-13

Total Pages: 358

ISBN-13: 1470437732

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This book is motivated by the problem of determining the set of rational points on a variety, but its true goal is to equip readers with a broad range of tools essential for current research in algebraic geometry and number theory. The book is unconventional in that it provides concise accounts of many topics instead of a comprehensive account of just one—this is intentionally designed to bring readers up to speed rapidly. Among the topics included are Brauer groups, faithfully flat descent, algebraic groups, torsors, étale and fppf cohomology, the Weil conjectures, and the Brauer-Manin and descent obstructions. A final chapter applies all these to study the arithmetic of surfaces. The down-to-earth explanations and the over 100 exercises make the book suitable for use as a graduate-level textbook, but even experts will appreciate having a single source covering many aspects of geometry over an unrestricted ground field and containing some material that cannot be found elsewhere.

Many Rational Points
Many Rational Points

Author: N.E. Hurt

Publisher: Springer Science & Business Media

Published: 2013-11-11

Total Pages: 368

ISBN-13: 9401702519

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This volume provides a source book of examples with relationships to advanced topics regarding Sato-Tate conjectures, Eichler-Selberg trace formula, Katz-Sarnak conjectures and Hecke operators." "The book will be of use to mathematicians, physicists and engineers interested in the mathematical methods of algebraic geometry as they apply to coding theory and cryptography."--Jacket

Rational Points on Modular Elliptic Curves
Rational Points on Modular Elliptic Curves

Author: Henri Darmon

Publisher: American Mathematical Soc.

Published: 2004

Total Pages: 146

ISBN-13: 0821828681

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The book surveys some recent developments in the arithmetic of modular elliptic curves. It places a special emphasis on the construction of rational points on elliptic curves, the Birch and Swinnerton-Dyer conjecture, and the crucial role played by modularity in shedding light on these two closely related issues. The main theme of the book is the theory of complex multiplication, Heegner points, and some conjectural variants. The first three chapters introduce the background and prerequisites: elliptic curves, modular forms and the Shimura-Taniyama-Weil conjecture, complex multiplication and the Heegner point construction. The next three chapters introduce variants of modular parametrizations in which modular curves are replaced by Shimura curves attached to certain indefinite quaternion algebras. The main new contributions are found in Chapters 7-9, which survey the author's attempts to extend the theory of Heegner points and complex multiplication to situations where the base field is not a CM field. Chapter 10 explains the proof of Kolyvagin's theorem, which relates Heegner points to the arithmetic of elliptic curves and leads to the best evidence so far for the Birch and Swinnerton-Dyer conjecture.

Rational Points on Algebraic Varieties
Rational Points on Algebraic Varieties

Author: Emmanuel Peyre

Publisher: Birkhäuser

Published: 2012-12-06

Total Pages: 455

ISBN-13: 3034883684

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This book is devoted to the study of rational and integral points on higher-dimensional algebraic varieties. It contains carefully selected research papers addressing the arithmetic geometry of varieties which are not of general type, with an emphasis on how rational points are distributed with respect to the classical, Zariski and adelic topologies. The present volume gives a glimpse of the state of the art of this rapidly expanding domain in arithmetic geometry. The techniques involve explicit geometric constructions, ideas from the minimal model program in algebraic geometry as well as analytic number theory and harmonic analysis on adelic groups.

Rational Points on Curves Over Finite Fields
Rational Points on Curves Over Finite Fields

Author: Harald Niederreiter

Publisher: Cambridge University Press

Published: 2001-06-14

Total Pages: 260

ISBN-13: 9780521665438

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Ever since the seminal work of Goppa on algebraic-geometry codes, rational points on algebraic curves over finite fields have been an important research topic for algebraic geometers and coding theorists. The focus in this application of algebraic geometry to coding theory is on algebraic curves over finite fields with many rational points (relative to the genus). Recently, the authors discovered another important application of such curves, namely to the construction of low-discrepancy sequences. These sequences are needed for numerical methods in areas as diverse as computational physics and mathematical finance. This has given additional impetus to the theory of, and the search for, algebraic curves over finite fields with many rational points. This book aims to sum up the theoretical work on algebraic curves over finite fields with many rational points and to discuss the applications of such curves to algebraic coding theory and the construction of low-discrepancy sequences.

Diophantine Geometry
Diophantine Geometry

Author: Marc Hindry

Publisher: Springer Science & Business Media

Published: 2013-12-01

Total Pages: 574

ISBN-13: 1461212103

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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.

Arakelov Geometry and Diophantine Applications
Arakelov Geometry and Diophantine Applications

Author: Emmanuel Peyre

Publisher: Springer Nature

Published: 2021-03-10

Total Pages: 469

ISBN-13: 3030575594

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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.

Elementary Theory of Numbers
Elementary Theory of Numbers

Author: W. Sierpinski

Publisher: Elsevier

Published: 1988-02-01

Total Pages: 527

ISBN-13: 0080960197

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Since the publication of the first edition of this work, considerable progress has been made in many of the questions examined. This edition has been updated and enlarged, and the bibliography has been revised.The variety of topics covered here includes divisibility, diophantine equations, prime numbers (especially Mersenne and Fermat primes), the basic arithmetic functions, congruences, the quadratic reciprocity law, expansion of real numbers into decimal fractions, decomposition of integers into sums of powers, some other problems of the additive theory of numbers and the theory of Gaussian integers.

The Arithmetic of Elliptic Curves
The Arithmetic of Elliptic Curves

Author: Joseph H. Silverman

Publisher: Springer Science & Business Media

Published: 2013-03-09

Total Pages: 414

ISBN-13: 1475719205

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The theory of elliptic curves is distinguished by its long history and by the diversity of the methods that have been used in its study. This book treats the arithmetic approach in its modern formulation, through the use of basic algebraic number theory and algebraic geometry. Following a brief discussion of the necessary algebro-geometric results, the book proceeds with an exposition of the geometry and the formal group of elliptic curves, elliptic curves over finite fields, the complex numbers, local fields, and global fields. Final chapters deal with integral and rational points, including Siegels theorem and explicit computations for the curve Y = X + DX, while three appendices conclude the whole: Elliptic Curves in Characteristics 2 and 3, Group Cohomology, and an overview of more advanced topics.