The Arithmetic of Elliptic Curves

Author: Joseph H. Silverman

Publisher: Springer Science & Business Media

ISBN: 9780387094946

Category: Mathematics

Page: 513

View: 1642

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.
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Advanced Topics in the Arithmetic of Elliptic Curves

Author: Joseph H. Silverman

Publisher: Springer Science & Business

ISBN: 9780387943251

Category: Mathematics

Page: 525

View: 8254

In "The Arithmetic of Elliptic Curves," the author presented the basic theory culminating in two fundamental global results, the Mordell-Weil theorem on the finite generation of the group of rational points and Siegel's theorem on the finiteness of the set of integral points. This book continues the study of elliptic curves by presenting six important, but somewhat more specialized topics: I. Elliptic and modular functions for the full modular group. II. Elliptic curves with complex multiplication. III. Elliptic surfaces and specialization theorems. IV. NA(c)ron models, Kodaira-N ron classification of special fibres, Tate's algorithm, and Ogg's conductor-discriminant formula. V. Tate's theory of q-curves over p-adic fields. VI. NA(c)ron's theory of canonical local height functions.
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Rational Points on Elliptic Curves

Author: Joseph H. Silverman,John T. Tate

Publisher: Springer

ISBN: 3319185888

Category: Mathematics

Page: 332

View: 7321

The theory of elliptic curves involves a pleasing blend of algebra, geometry, analysis, and number theory. This volume stresses this interplay as it develops the basic theory, thereby providing an opportunity for advanced undergraduates to appreciate the unity of modern mathematics. At the same time, every effort has been made to use only methods and results commonly included in the undergraduate curriculum. This accessibility, the informal writing style, and a wealth of exercises make Rational Points on Elliptic Curves an ideal introduction for students at all levels who are interested in learning about Diophantine equations and arithmetic geometry. Most concretely, an elliptic curve is the set of zeroes of a cubic polynomial in two variables. If the polynomial has rational coefficients, then one can ask for a description of those zeroes whose coordinates are either integers or rational numbers. It is this number theoretic question that is the main subject of Rational Points on Elliptic Curves. Topics covered include the geometry and group structure of elliptic curves, the Nagell–Lutz theorem describing points of finite order, the Mordell–Weil theorem on the finite generation of the group of rational points, the Thue–Siegel theorem on the finiteness of the set of integer points, theorems on counting points with coordinates in finite fields, Lenstra's elliptic curve factorization algorithm, and a discussion of complex multiplication and the Galois representations associated to torsion points. Additional topics new to the second edition include an introduction to elliptic curve cryptography and a brief discussion of the stunning proof of Fermat's Last Theorem by Wiles et al. via the use of elliptic curves.
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Elliptic Curves

Author: Dale Husemoller

Publisher: Springer Science & Business Media

ISBN: 1475751192

Category: Mathematics

Page: 350

View: 7327

The book divides naturally into several parts according to the level of the material, the background required of the reader, and the style of presentation with respect to details of proofs. For example, the first part, to Chapter 6, is undergraduate in level, the second part requires a background in Galois theory and the third some complex analysis, while the last parts, from Chapter 12 on, are mostly at graduate level. A general outline ofmuch ofthe material can be found in Tate's colloquium lectures reproduced as an article in Inven tiones [1974]. The first part grew out of Tate's 1961 Haverford Philips Lectures as an attempt to write something for publication c10sely related to the original Tate notes which were more or less taken from the tape recording of the lectures themselves. This inc1udes parts of the Introduction and the first six chapters The aim ofthis part is to prove, by elementary methods, the Mordell theorem on the finite generation of the rational points on elliptic curves defined over the rational numbers. In 1970 Tate teturned to Haverford to give again, in revised form, the originallectures of 1961 and to extend the material so that it would be suitable for publication. This led to a broader plan forthe book.
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Elliptic Curves

A Computational Approach

Author: Susanne Schmitt,Horst G. Zimmer

Publisher: Walter de Gruyter

ISBN: 3110198010

Category: Mathematics

Page: 376

View: 5771

The basics of the theory of elliptic curves should be known to everybody, be he (or she) a mathematician or a computer scientist. Especially everybody concerned with cryptography should know the elements of this theory. The purpose of the present textbook is to give an elementary introduction to elliptic curves. Since this branch of number theory is particularly accessible to computer-assisted calculations, the authors make use of it by approaching the theory under a computational point of view. Specifically, the computer-algebra package SIMATH can be applied on several occasions. However, the book can be read also by those not interested in any computations. Of course, the theory of elliptic curves is very comprehensive and becomes correspondingly sophisticated. That is why the authors made a choice of the topics treated. Topics covered include the determination of torsion groups, computations regarding the Mordell-Weil group, height calculations, S-integral points. The contents is kept as elementary as possible. In this way it becomes obvious in which respect the book differs from the numerous textbooks on elliptic curves nowadays available.
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Ebene algebraische Kurven

Author: Egbert Brieskorn,Horst Knörrer

Publisher: N.A

ISBN: N.A

Category: Mathematics

Page: 964

View: 898

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Introduction to Elliptic Curves and Modular Forms

Author: Neal I. Koblitz

Publisher: Springer Science & Business Media

ISBN: 1461209099

Category: Mathematics

Page: 252

View: 3935

The theory of elliptic curves and modular forms provides a fruitful meeting ground for such diverse areas as number theory, complex analysis, algebraic geometry, and representation theory. This book starts out with a problem from elementary number theory and proceeds to lead its reader into the modern theory, covering such topics as the Hasse-Weil L-function and the conjecture of Birch and Swinnerton-Dyer. This new edition details the current state of knowledge of elliptic curves.
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Elliptic Tales

Curves, Counting, and Number Theory

Author: Avner Ash,Robert Gross

Publisher: Princeton University Press

ISBN: 1400841712

Category: Mathematics

Page: 280

View: 2777

Elliptic Tales describes the latest developments in number theory by looking at one of the most exciting unsolved problems in contemporary mathematics—the Birch and Swinnerton-Dyer Conjecture. In this book, Avner Ash and Robert Gross guide readers through the mathematics they need to understand this captivating problem. The key to the conjecture lies in elliptic curves, which may appear simple, but arise from some very deep—and often very mystifying—mathematical ideas. Using only basic algebra and calculus while presenting numerous eye-opening examples, Ash and Gross make these ideas accessible to general readers, and, in the process, venture to the very frontiers of modern mathematics.
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Algorithmic Number Theory

9th International Symposium, ANTS-IX, Nancy, France, July 19-23, 2010, Proceedings

Author: Guillaume Hanrot,Francois Morain,Emmanuel Thomé

Publisher: Springer Science & Business Media

ISBN: 3642145175

Category: Computers

Page: 397

View: 5277

This book constitutes the refereed proceedings of the 9th International Algorithmic Number Theory Symposium, ANTS 2010, held in Nancy, France, in July 2010. The 25 revised full papers presented together with 5 invited papers were carefully reviewed and selected for inclusion in the book. The papers are devoted to algorithmic aspects of number theory, including elementary number theory, algebraic number theory, analytic number theory, geometry of numbers, algebraic geometry, finite fields, and cryptography.
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Variations on a Theme of Euler

Author: Takashi Ono

Publisher: Springer Science & Business Media

ISBN: 9780306447891

Category: Mathematics

Page: 347

View: 9624

In this first-of-its-kind book, Professor Ono postulates that one aspect of classical and modern number theory, including quadratic forms and space elliptic curves as intersections of quadratic surfaces, can be considered as the number theory of Hopf maps. The text, a translation of Dr. Ono's earlier work, provides a solution to this problem by employing three areas of mathematics: linear algebra, algebraic geometry, and simple algebras. This English-language edition presents a new chapter on arithmetic of quadratic maps, along with an appendix featuring a short survey of subsequent research on congruent numbers by Masanari Kida. The original appendix containing historical and scientific comments on Euler's Elements of Algebra is also included. Variations on a Theme of Euler is an important reference for researchers and an excellent text for a graduate-level course on number theory.
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Revue de mathématique élémentaires

Author: N.A

Publisher: N.A

ISBN: N.A

Category: Mathematics

Page: N.A

View: 5743

Elemente der Mathematik (EL) publishes survey articles about important developments in the field of mathematics; stimulating shorter communications that tackle more specialized questions; and papers that report on the latest advances in mathematics and applications in other disciplines. The journal does not focus on basic research. Rather, its articles seek to convey to a wide circle of readers - teachers, students, engineers, professionals in industry and administration - the relevance, intellectual challenge and vitality of mathematics today. The Problems Section, covering a diverse range of exercises of varying degrees of difficulty, encourages an active grappling with mathematical problems.
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Elliptic Curves, Modular Forms & Fermat's Last Theorem

Proceedings of a Conference Held in the Institute of Mathematics of the Chinese University of Hong Kong

Author: John Coates,Shing-Tung Yau

Publisher: International Pressof Boston Incorporated

ISBN: 9781571460493

Category: Mathematics

Page: 340

View: 4040

These proceedings are based on a conference at the Chinese University of Hong Kong, held in response to Andrew Wile's conjecture that every elliptic curve over Q is modular. The survey article describing Wile's work is included as the first article in the present edition.
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LMSST: 24 Lectures on Elliptic Curves

Author: John William Scott Cassels

Publisher: Cambridge University Press

ISBN: 9780521425308

Category: Mathematics

Page: 137

View: 719

The study of special cases of elliptic curves goes back to Diophantos and Fermat, and today it is still one of the liveliest centers of research in number theory. This book, addressed to beginning graduate students, introduces basic theory from a contemporary viewpoint but with an eye to the historical background. The central portion deals with curves over the rationals: the Mordell-Wei finite basis theorem, points of finite order (Nagell-Lutz), etc. The treatment is structured by the local-global standpoint and culminates in the description of the Tate-Shafarevich group as the obstruction to a Hasse principle. In an introductory section the Hasse principle for conics is discussed. The book closes with sections on the theory over finite fields (the "Riemann hypothesis for function fields") and recently developed uses of elliptic curves for factoring large integers. Prerequisites are kept to a minimum; an acquaintance with the fundamentals of Galois theory is assumed, but no knowledge either of algebraic number theory or algebraic geometry is needed. The p-adic numbers are introduced from scratch. Many examples and exercises are included for the reader, and those new to elliptic curves, whether they are graduate students or specialists from other fields, will find this a valuable introduction.
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Algebraic Function

Fields and Codes

Author: Henning Stichtenoth

Publisher: Springer Science & Business Media

ISBN: 9783540564898

Category: Mathematics

Page: 260

View: 1599

This book has two objectives. The first is to fill a void in the existing mathematical literature by providing a modern, self-contained and in-depth exposition of the theory of algebraic function fields. Topics include the Riemann-Roch theorem, algebraic extensions of function fields, ramifications theory and differentials. Particular emphasis is placed on function fields over a finite constant field, leading into zeta functions and the Hasse-Weil theorem. Numerous examples illustrate the general theory. Error-correcting codes are in widespread use for the reliable transmission of information. Perhaps the most fascinating of all the ties that link the theory of these codes to mathematics is the construction by V.D. Goppa, of powerful codes using techniques borrowed from algebraic geometry. Algebraic function fields provide the most elementary approach to Goppa's ideas, and the second objective of this book is to provide an introduction to Goppa's algebraic-geometric codes along these lines. The codes, their parameters and links with traditional codes such as classical Goppa, Peed-Solomon and BCH codes are treated at an early stage of the book. Subsequent chapters include a decoding algorithm for these codes as well as a discussion of their subfield subcodes and trace codes. Stichtenoth's book will be very useful to students and researchers in algebraic geometry and coding theory and to computer scientists and engineers interested in information transmission.
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Buchführung 1 DATEV-Kontenrahmen 2018

Grundlagen der Buchführung für Industrie- und Handelsbetriebe

Author: Manfred Bornhofen,Martin C. Bornhofen

Publisher: Springer-Verlag

ISBN: 3658216948

Category: Business & Economics

Page: 476

View: 3417

Der vorliegende Band Buchführung 1 bietet Ihnen in bewährter Didaktik einen schnellen und leicht verständlichen Zugang zu den Grundlagen der Buchführung. Der an den Anforderungen der Praxis ausgerichtete Aufgabenteil umfasst Aufgaben mit unterschiedlichem Schwierigkeitsgrad. Weitere Aufgaben und Lösungen zur Verstärkung des Lernerfolgs enthält das zur Buchführung 1 erhältliche Lösungsbuch. – Dem Werk liegen die in der Praxis am häufigsten verwendeten DATEV-Kontenrahmen SKR 04 und SKR 03 zugrunde. Sie sind kompatibel mit den wichtigsten übrigen Kontenrahmen (z. B. GKR und IKR). – Die 30., überarbeitete Auflage berücksichtigt die bis zum 31.05.2018 maßgebliche Rechtslage, insbesondere das Gesetz gegen schädliche Steuerpraktiken im Zusammenhang mit Rechteüberlassungen, das zweite Bürokratieentlastungsgesetz sowie aktuelle BMF-Schreiben und sonstige Änderungen. – Rechtsänderungen, die sich ab 01.06.2018 noch für 2018 ergeben, können Sie kostenlos als „Online Plus“ auf der Homepage zum Buch abrufen. Damit wird der komplette Rechtsstand für das Jahr 2018 garantiert. Ihr zusätzlicher Mehrwert: eBook inside! Die gesamte Bornhofen Edition erscheint mit eBook inside, um das digitale Arbeiten mit dem Unterrichts- und Lernstoff zu erleichtern – ein relevanter Mehrwert für alle Lehrenden und Lernenden. Ausgewählte Verlinkungen zu Gesetzestexten, BMF-Schreiben u. a. ermöglichen ein innovatives Lernerlebnis, das analoge und digitale Inhalte praxisrelevant miteinander verknüpft. Begleitend zum Lehrbuch ist auch ein Lösungsbuch mit weiteren Prüfungsaufgaben und Lösungen zur Rechtslage des Jahres 2018 (ISBN 978-3-658-21695-5) erhältlich.
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Galois Cohomology of Elliptic Curves

Author: John Coates,R. Sujatha

Publisher: Alpha Science International, Limited

ISBN: N.A

Category: Mathematics

Page: 100

View: 2999

This book is based on the material presented in four lectures given by J. Coates at the Tata Institute of Fundamental Research. The original notes were modified and expanded in a joint project with R. Sujatha. The book discusses some aspects of the Iwasawa theory of elliptic curves over algebraic fields. Let E be an elliptic curve defined over an algebraic number field F. The fundamental idea of the Iwasawa theory is to study deep arithmetic questions about E/F, via the study of coarser questions about the arithmetic of E over various infinite extensions of F.
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