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Elliptic Curves

Elliptic Curves Author Henry McKean
ISBN-10 0521658179
Release 1999-08-13
Pages 280
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The subject of elliptic curves is one of the jewels of nineteenth-century mathematics, originated by Abel, Gauss, Jacobi, and Legendre. This 1997 book presents an introductory account of the subject in the style of the original discoverers, with references to and comments about more recent and modern developments. It combines three of the fundamental themes of mathematics: complex function theory, geometry, and arithmetic. After an informal preparatory chapter, the book follows an historical path, beginning with the work of Abel and Gauss on elliptic integrals and elliptic functions. This is followed by chapters on theta functions, modular groups and modular functions, the quintic, the imaginary quadratic field, and on elliptic curves. Requiring only a first acquaintance with complex function theory, this book is an ideal introduction to the subject for graduate students and researchers in mathematics and physics, with many exercises with hints scattered throughout the text.

The Arithmetic of Elliptic Curves

The Arithmetic of Elliptic Curves Author Joseph H. Silverman
ISBN-10 9781475719208
Release 2013-03-09
Pages 402
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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.

Elliptic Curves MN 40

Elliptic Curves   MN 40 Author Anthony W. Knapp
ISBN-10 9780691186900
Release 2018-06-05
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Elliptic Curves MN 40 has been writing in one form or another for most of life. You can find so many inspiration from Elliptic Curves MN 40 also informative, and entertaining. Click DOWNLOAD or Read Online button to get full Elliptic Curves MN 40 book for free.

Advanced Topics in the Arithmetic of Elliptic Curves

Advanced Topics in the Arithmetic of Elliptic Curves Author Joseph H. Silverman
ISBN-10 9781461208518
Release 2013-12-01
Pages 528
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In the introduction to the first volume of The Arithmetic of Elliptic Curves (Springer-Verlag, 1986), I observed that "the theory of elliptic curves is rich, varied, and amazingly vast," and as a consequence, "many important topics had to be omitted." I included a brief introduction to ten additional topics as an appendix to the first volume, with the tacit understanding that eventually there might be a second volume containing the details. You are now holding that second volume. it turned out that even those ten topics would not fit Unfortunately, into a single book, so I was forced to make some choices. The following material is covered in this book: I. Elliptic and modular functions for the full modular group. II. Elliptic curves with complex multiplication. III. Elliptic surfaces and specialization theorems. IV. Neron models, Kodaira-Neron classification of special fibers, Tate's algorithm, and Ogg's conductor-discriminant formula. V. Tate's theory of q-curves over p-adic fields. VI. Neron's theory of canonical local height functions.

Rational Points on Elliptic Curves

Rational Points on Elliptic Curves Author Joseph H. Silverman
ISBN-10 0387978259
Release 1992-06-24
Pages 281
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In 1961 the second author deliv1lred a series of lectures at Haverford Col lege on the subject of "Rational Points on Cubic Curves. " These lectures, intended for junior and senior mathematics majors, were recorded, tran scribed, and printed in mimeograph form. Since that time they have been widely distributed as photocopies of ever decreasing legibility, and por tions have appeared in various textbooks (Husemoller [1], Chahal [1]), but they have never appeared in their entirety. In view of the recent inter est in the theory of elliptic curves for subjects ranging from cryptogra phy (Lenstra [1], Koblitz [2]) to physics (Luck-Moussa-Waldschmidt [1]), as well as the tremendous purely mathematical activity in this area, it seems a propitious time to publish an expanded version of those original notes suitable for presentation to an advanced undergraduate audience. We have attempted to maintain much of the informality of the orig inal Haverford lectures. Our main goal in doing this has been to write a textbook in a technically difficult field which is "readable" by the average undergraduate mathematics major. We hope we have succeeded in this goal. The most obvious drawback to such an approach is that we have not been entirely rigorous in all of our proofs. In particular, much of the foundational material on elliptic curves presented in Chapter I is meant to explain and convince, rather than to rigorously prove.

Elliptic Curves

Elliptic Curves Author Dale Husemöller
ISBN-10 9780387954905
Release 2004
Pages 487
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This book is an introduction to the theory of elliptic curves, ranging from its elementary aspects to current research. This new edition contains three new chapters which explore recent directions and extensions of the theory and two new appendices: the first, by Stefan Theisan, examines the role of Calabi-Yau manifolds in string theory while the second, by Otto Forster, discusses the use of elliptic curves in computing theory and coding theory.

Elliptic Functions and Elliptic Integrals

Elliptic Functions and Elliptic Integrals Author Viktor Vasil_evich Prasolov
ISBN-10 0821897802
Release 1997-09-16
Pages 185
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This book is devoted to the geometry and arithmetic of elliptic curves and to elliptic functions with applications to algebra and number theory. It includes modern interpretations of some famous classical algebraic theorems such as Abel's theorem on the lemniscate and Hermite's solution of the fifth degree equation by means of theta functions. Suitable as a text, the book is self-contained and assumes as prerequisites only the standard one-year courses of algebra and analysis.

Using the Mathematics Literature

Using the Mathematics Literature Author Kristine K. Fowler
ISBN-10 0824750357
Release 2004-05-25
Pages 475
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This reference serves as a reader-friendly guide to every basic tool and skill required in the mathematical library and helps mathematicians find resources in any format in the mathematics literature. It lists a wide range of standard texts, journals, review articles, newsgroups, and Internet and database tools for every major subfield in mathematics and details methods of access to primary literature sources of new research, applications, results, and techniques. Using the Mathematics Literature is the most comprehensive and up-to-date resource on mathematics literature in both print and electronic formats, presenting time-saving strategies for retrieval of the latest information.

Higher Regulators Algebraic K Theory and Zeta Functions of Elliptic Curves

Higher Regulators  Algebraic K Theory  and Zeta Functions of Elliptic Curves Author Spencer J. Bloch
ISBN-10 9780821829738
Release 2000
Pages 97
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This brief hardcover is a classic title covering a renowned series of lectures. A mathematical jewel.

Elliptic Curves

Elliptic Curves Author Lawrence C. Washington
ISBN-10 1420071475
Release 2008-04-03
Pages 536
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Like its bestselling predecessor, Elliptic Curves: Number Theory and Cryptography, Second Edition develops the theory of elliptic curves to provide a basis for both number theoretic and cryptographic applications. With additional exercises, this edition offers more comprehensive coverage of the fundamental theory, techniques, and applications of elliptic curves. New to the Second Edition Chapters on isogenies and hyperelliptic curves A discussion of alternative coordinate systems, such as projective, Jacobian, and Edwards coordinates, along with related computational issues A more complete treatment of the Weil and Tate–Lichtenbaum pairings Doud’s analytic method for computing torsion on elliptic curves over Q An explanation of how to perform calculations with elliptic curves in several popular computer algebra systems Taking a basic approach to elliptic curves, this accessible book prepares readers to tackle more advanced problems in the field. It introduces elliptic curves over finite fields early in the text, before moving on to interesting applications, such as cryptography, factoring, and primality testing. The book also discusses the use of elliptic curves in Fermat’s Last Theorem. Relevant abstract algebra material on group theory and fields can be found in the appendices.

Elliptic Curves and Arithmetic Invariants

Elliptic Curves and Arithmetic Invariants Author Haruzo Hida
ISBN-10 9781461466574
Release 2013-06-13
Pages 450
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This book contains a detailed account of the result of the author's recent Annals paper and JAMS paper on arithmetic invariant, including μ-invariant, L-invariant, and similar topics. This book can be regarded as an introductory text to the author's previous book p-Adic Automorphic Forms on Shimura Varieties. Written as a down-to-earth introduction to Shimura varieties, this text includes many examples and applications of the theory that provide motivation for the reader. Since it is limited to modular curves and the corresponding Shimura varieties, this book is not only a great resource for experts in the field, but it is also accessible to advanced graduate students studying number theory. Key topics include non-triviality of arithmetic invariants and special values of L-functions; elliptic curves over complex and p-adic fields; Hecke algebras; scheme theory; elliptic and modular curves over rings; and Shimura curves.

Algorithmic Arithmetic Geometry and Coding Theory

Algorithmic Arithmetic  Geometry  and Coding Theory Author Stéphane Ballet
ISBN-10 9781470414610
Release 2015-04-20
Pages 306
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This volume contains the proceedings of the 14th International Conference on Arithmetic, Geometry, Cryptography, and Coding Theory (AGCT), held June 3-7, 2013, at CIRM, Marseille, France. These international conferences, held every two years, have been a major event in the area of algorithmic and applied arithmetic geometry for more than 20 years. This volume contains 13 original research articles covering geometric error correcting codes, and algorithmic and explicit arithmetic geometry of curves and higher dimensional varieties. Tools used in these articles include classical algebraic geometry of curves, varieties and Jacobians, Suslin homology, Monsky-Washnitzer cohomology, and -functions of modular forms.

Mathematical Tools for Physicists

Mathematical Tools for Physicists Author Michael Grinfeld
ISBN-10 9783527684274
Release 2014-11-05
Pages 632
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The new edition is significantly updated and expanded. This unique collection of review articles, ranging from fundamental concepts up to latest applications, contains individual contributions written by renowned experts in the relevant fields. Much attention is paid to ensuring fast access to the information, with each carefully reviewed article featuring cross-referencing, references to the most relevant publications in the field, and suggestions for further reading, both introductory as well as more specialized. While the chapters on group theory, integral transforms, Monte Carlo methods, numerical analysis, perturbation theory, and special functions are thoroughly rewritten, completely new content includes sections on commutative algebra, computational algebraic topology, differential geometry, dynamical systems, functional analysis, graph and network theory, PDEs of mathematical physics, probability theory, stochastic differential equations, and variational methods.

Arithmetic Theory of Elliptic Curves

Arithmetic Theory of Elliptic Curves Author J. Coates
ISBN-10 9783540481607
Release 2006-11-14
Pages 264
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This volume contains the expanded versions of the lectures given by the authors at the C.I.M.E. instructional conference held in Cetraro, Italy, from July 12 to 19, 1997. The papers collected here are broad surveys of the current research in the arithmetic of elliptic curves, and also contain several new results which cannot be found elsewhere in the literature. Owing to clarity and elegance of exposition, and to the background material explicitly included in the text or quoted in the references, the volume is well suited to research students as well as to senior mathematicians.

Modern Cryptography and Elliptic Curves A Beginner s Guide

Modern Cryptography and Elliptic Curves  A Beginner   s Guide Author Thomas R. Shemanske
ISBN-10 9781470435820
Release 2017-07-31
Pages 252
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This book offers the beginning undergraduate student some of the vista of modern mathematics by developing and presenting the tools needed to gain an understanding of the arithmetic of elliptic curves over finite fields and their applications to modern cryptography. This gradual introduction also makes a significant effort to teach students how to produce or discover a proof by presenting mathematics as an exploration, and at the same time, it provides the necessary mathematical underpinnings to investigate the practical and implementation side of elliptic curve cryptography (ECC). Elements of abstract algebra, number theory, and affine and projective geometry are introduced and developed, and their interplay is exploited. Algebra and geometry combine to characterize congruent numbers via rational points on the unit circle, and group law for the set of points on an elliptic curve arises from geometric intuition provided by Bézout's theorem as well as the construction of projective space. The structure of the unit group of the integers modulo a prime explains RSA encryption, Pollard's method of factorization, Diffie–Hellman key exchange, and ElGamal encryption, while the group of points of an elliptic curve over a finite field motivates Lenstra's elliptic curve factorization method and ECC. The only real prerequisite for this book is a course on one-variable calculus; other necessary mathematical topics are introduced on-the-fly. Numerous exercises further guide the exploration.

Arithmetic Moduli of Elliptic Curves

Arithmetic Moduli of Elliptic Curves Author Nicholas M. Katz
ISBN-10 0691083525
Release 1985-01-01
Pages 514
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This work is a comprehensive treatment of recent developments in the study of elliptic curves and their moduli spaces. The arithmetic study of the moduli spaces began with Jacobi's "Fundamenta Nova" in 1829, and the modern theory was erected by Eichler-Shimura, Igusa, and Deligne-Rapoport. In the past decade mathematicians have made further substantial progress in the field. This book gives a complete account of that progress, including not only the work of the authors, but also that of Deligne and Drinfeld.

Arithmetic Geometry

Arithmetic Geometry Author Nancy Childress
ISBN-10 9780821851746
Release 1994
Pages 220
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This book resulted from a research conference in arithmetic geometry held at Arizona State University in March 1993. The papers describe important recent advances in arithmetic geometry. Several articles deal with $p$-adic modular forms of half-integral weight and their roles in arithmetic geometry. The volume also contains material on the Iwasawa theory of cyclotomic fields, elliptic curves, and function fields, including $p$-adic $L$-functions and $p$-adic height pairings. Other articles focus on the inverse Galois problem, fields of definition of abelian varieties with real multiplication, and computation of torsion groups of elliptic curves. The volume also contains a previously unpublished letter of John Tate, written to J.-P. Serre in 1973, concerning Serre's conjecture on Galois representations. With contributions by some of the leading experts in the field, this book provides a look at the state of the art in arithmetic geometry.