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Undergraduate Texts in Mathematics

Rational Points on Elliptic Curves

Authors: Silverman, Joseph H., Tate, John T.

  • Written by leading experts in the field
  • Presents a technically difficult field which is readable by the average undergraduate mathematics major
  • Brings the classic first edition up-to-date with the inclusion of hot topics such as elliptic curve cryptography and the ABC Conjecture
  • Explains the role of Frey curves and Galois representations in Andrew Wiles' breakthrough proof of Fermat's Last Theorem
  • Includes many new exercises
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Buy this book

eBook $44.99
price for USA (gross)
  • ISBN 978-3-319-18588-0
  • Digitally watermarked, DRM-free
  • Included format: PDF, EPUB
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $59.99
price for USA
  • ISBN 978-3-319-18587-3
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $59.99
price for USA
  • Customers within the U.S. and Canada please contact Customer Service at 1-800-777-4643, Latin America please contact us at +1-212-460-1500 (Weekdays 8:30am – 5:30pm ET) to place your order.
  • Due: June 3, 2015
  • ISBN 978-3-319-30757-2
  • Free shipping for individuals worldwide
About this Textbook

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.

About the authors

Joseph H. Silverman is Professor of Mathematics at Brown University. He is the author of over 100 research articles and numerous books on elliptic curves, diophantine geometry, cryptography, and arithmetic dynamical systems.

John T. Tate is Professor Emeritus of Mathematics at The University of Texas at Austin and at Harvard University. For his seminal contributions to number theory, he was awarded the 2010 Abel Prize.

Reviews

“The two main changes for this edition are a new section on elliptic curve cryptography and an explanation of how elliptic curves played a role in the proof of Fermat’s Last Theorem. … the best place to start learning about elliptic curves.” (Fernando Q. Gouvêa, MAA Reviews, maa.org, April, 2016)

“The book is an excellent introduction to elliptic curves over the rational numbers and ideal textbook for an undergraduate course. … This book is highly recommended to students and researches interested in elliptic curves and their applications. It provides a natural step to a more advanced treatment of the subject.” (Andrej Dujella, zbMATH 1346.11001, 2016)


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Table of contents (6 chapters)

  • Geometry and Arithmetic

    Silverman, Joseph H. (et al.)

    Pages 1-34

  • Points of Finite Order

    Silverman, Joseph H. (et al.)

    Pages 35-63

  • The Group of Rational Points

    Silverman, Joseph H. (et al.)

    Pages 65-115

  • Cubic Curves over Finite Fields

    Silverman, Joseph H. (et al.)

    Pages 117-166

  • Integer Points on Cubic Curves

    Silverman, Joseph H. (et al.)

    Pages 167-205

Buy this book

eBook $44.99
price for USA (gross)
  • ISBN 978-3-319-18588-0
  • Digitally watermarked, DRM-free
  • Included format: PDF, EPUB
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $59.99
price for USA
  • ISBN 978-3-319-18587-3
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $59.99
price for USA
  • Customers within the U.S. and Canada please contact Customer Service at 1-800-777-4643, Latin America please contact us at +1-212-460-1500 (Weekdays 8:30am – 5:30pm ET) to place your order.
  • Due: June 3, 2015
  • ISBN 978-3-319-30757-2
  • Free shipping for individuals worldwide
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Bibliographic Information

Bibliographic Information
Book Title
Rational Points on Elliptic Curves
Authors
Series Title
Undergraduate Texts in Mathematics
Copyright
2015
Publisher
Springer International Publishing
Copyright Holder
Springer International Publishing Switzerland
eBook ISBN
978-3-319-18588-0
DOI
10.1007/978-3-319-18588-0
Hardcover ISBN
978-3-319-18587-3
Softcover ISBN
978-3-319-30757-2
Series ISSN
0172-6056
Edition Number
2
Number of Pages
XXII, 332
Number of Illustrations and Tables
37 b/w illustrations
Topics