# Rational Points on Elliptic Curves

• Joseph H. Silverman
• John T. Tate
Textbook

Part of the Undergraduate Texts in Mathematics book series (UTM)

1. Front Matter
Pages i-xxii
2. Joseph H. Silverman, John T. Tate
Pages 1-34
3. Joseph H. Silverman, John T. Tate
Pages 35-63
4. Joseph H. Silverman, John T. Tate
Pages 65-115
5. Joseph H. Silverman, John T. Tate
Pages 117-166
6. Joseph H. Silverman, John T. Tate
Pages 167-205
7. Joseph H. Silverman, John T. Tate
Pages 207-264
8. Back Matter
Pages 265-332

### Introduction

The theory of elliptic curves involves a pleasing blend of algebra, geometry, analysis, and number theory. This book 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 this book. 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.

### Keywords

ABC conjecture Fermat's last theorem Frey curves complex multiplication elliptic curve cryptography elliptic curves rational points

#### Authors and affiliations

• Joseph H. Silverman
• 1
• John T. Tate
• 2
1. 1.MathematicsBrown UniversityProvidenceUSA
2. 2.Dept. of MathematicsHarvard UniversityCambridgeUSA

### Bibliographic information

• DOI https://doi.org/10.1007/978-3-319-18588-0
• Copyright Information Springer International Publishing Switzerland 2015
• Publisher Name Springer, Cham
• eBook Packages Mathematics and Statistics
• Print ISBN 978-3-319-18587-3
• Online ISBN 978-3-319-18588-0
• Series Print ISSN 0172-6056
• Series Online ISSN 2197-5604
• Buy this book on publisher's site