Elliptic Tales: Curves, Counting, and Number Theory

Hardcover (Print)
Used and New from Other Sellers
Used and New from Other Sellers
from $14.98
Usually ships in 1-2 business days
(Save 49%)
Other sellers (Hardcover)
  • All (16) from $14.98   
  • New (9) from $17.54   
  • Used (7) from $14.98   

Overview

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. The Clay Mathematics Institute is offering a prize of $1 million to anyone who can discover a general solution to the problem. 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 are cubic equations in two variables. These equations may appear simple, yet they 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. Along the way, they give an informative and entertaining introduction to some of the most profound discoveries of the last three centuries in algebraic geometry, abstract algebra, and number theory. They demonstrate how mathematics grows more abstract to tackle ever more challenging problems, and how each new generation of mathematicians builds on the accomplishments of those who preceded them. Ash and Gross fully explain how the Birch and Swinnerton-Dyer Conjecture sheds light on the number theory of elliptic curves, and how it provides a beautiful and startling connection between two very different objects arising from an elliptic curve, one based on calculus, the other on algebra.

Read More Show Less

Editorial Reviews

Mathematics Magazine
Minimal prerequisites and its clear writing make this book (which even has a few exercises) a great choice for a seminar for mathematics majors, who at some point should have such an excursion to one of the frontiers of mathematics.
Times Higher Education
The authors of Elliptic Tales do a superb job in demonstrating the approach that mathematicians take when they confront unsolved problems involving elliptic curves.
— Sungkon Chang
SIAM News
One cannot help being impressed, in reading the book and pursuing a few of the references, by the magnitude of the enterprise it chronicles.
— James Case
Choice
Ash and Gross thoroughly explain the statement and significance of the linchpin Birch and Swinnerton-Dyer conjection. . . . [A]sh and Gross deliver ample and current intellectual and technical substance.
Times Higher Education - Sungkon Chang
The authors of Elliptic Tales do a superb job in demonstrating the approach that mathematicians take when they confront unsolved problems involving elliptic curves.
SIAM News - James Case
One cannot help being impressed, in reading the book and pursuing a few of the references, by the magnitude of the enterprise it chronicles.
Mathematical Reviews Clippings - Lisa A. Berger
I would envision this book as an excellent text for an undergraduate 'capstone' course in mathematics; the book lends itself to independent reading, but topics may be explored in much greater depth and rigor in the classroom. Additionally, the book indeed brings together ideas from calculus, complex variables and algebra, showing how a single mathematical research question may require an integrated understanding of the various branches of mathematics. Thus, it encourages students to reinforce their understanding of these various fields, while simultaneously introducing them to an open question in mathematics and a vibrant field of study.
From the Publisher
"The authors present their discussion in an informal, sometimes playful manner and with detail that will appeal to an audience with a basic understanding of calculus. This book will captivate math enthusiasts as well as readers curious about an intriguing and still unanswered question."—Margaret Dominy, Library Journal

"Minimal prerequisites and its clear writing make this book (which even has a few exercises) a great choice for a seminar for mathematics majors, who at some point should have such an excursion to one of the frontiers of mathematics."Mathematics Magazine

"The authors of Elliptic Tales do a superb job in demonstrating the approach that mathematicians take when they confront unsolved problems involving elliptic curves."—Sungkon Chang, Times Higher Education

"One cannot help being impressed, in reading the book and pursuing a few of the references, by the magnitude of the enterprise it chronicles."—James Case, SIAM News

"Ash and Gross thoroughly explain the statement and significance of the linchpin Birch and Swinnerton-Dyer conjection. . . . [A]sh and Gross deliver ample and current intellectual and technical substance."Choice

"I would envision this book as an excellent text for an undergraduate 'capstone' course in mathematics; the book lends itself to independent reading, but topics may be explored in much greater depth and rigor in the classroom. Additionally, the book indeed brings together ideas from calculus, complex variables and algebra, showing how a single mathematical research question may require an integrated understanding of the various branches of mathematics. Thus, it encourages students to reinforce their understanding of these various fields, while simultaneously introducing them to an open question in mathematics and a vibrant field of study."—Lisa A. Berger, Mathematical Reviews Clippings

"The book is very pleasantly written, and in my opinion, the authors have done an admirable job in giving an idea to non-experts what the Birch-Swinnerton Dyer conjecture is about."—Jan-Hendrik Evertse, Zentralblatt MATH

"The book's most important contributions . . . are the sense of discovery, invention, and insight into the habits of mind used by mathematicians on this journey. I would recommend this book to anyone who wants to be challenged mathematically or who wants to experience mathematics as creative and exciting."—Jacqueline Coomes, Mathematics Teacher

"[T]his book is a wonderful introduction to what is arguably one of the most important mathematical problems of our time and for that reason alone it deserves to be widely read. Another reason to recommend this book is the opportunity to share in the readily apparent joy the authors have for their subject and the beauty they see in it, not least because . . . joy and beauty are the most important reasons for doing mathematics, irrespective of its dollar value."—Rob Ashmore, Mathematics Today

Read More Show Less

Product Details

  • ISBN-13: 9780691151199
  • Publisher: Princeton University Press
  • Publication date: 3/12/2012
  • Pages: 280
  • Sales rank: 979,246
  • Product dimensions: 6.10 (w) x 9.20 (h) x 1.00 (d)

Meet the Author

Avner Ash is professor of mathematics at Boston College. Robert Gross is associate professor of mathematics at Boston College. They are the coauthors of "Fearless Symmetry: Exposing the Hidden Patterns of Numbers" (Princeton).

Read More Show Less

Table of Contents

Preface xiii
Acknowledgments xix
Prologue 1

PART I. DEGREE

Chapter 1. Degree of a Curve 13
1. Greek Mathematics 13
2. Degree 14
3. Parametric Equations 20
4. Our Two Definitions of Degree Clash 23

Chapter 2. Algebraic Closures 26
1. Square Roots of Minus One 26
2. Complex Arithmetic 28
3. Rings and Fields 30
4. Complex Numbers and Solving Equations 32
5. Congruences 34
6. Arithmetic Modulo a Prime 38
7. Algebraic Closure 38

Chapter 3. The Projective Plane 42
1. Points at Infinity 42
2. Projective Coordinates on a Line 46
3. Projective Coordinates on a Plane 50
4. Algebraic Curves and Points at Infinity 54
5. Homogenization of Projective Curves 56
6. Coordinate Patches 61

Chapter 4. Multiplicities and Degree 67
1. Curves as Varieties 67
2. Multiplicities 69
3. Intersection Multiplicities 72
4. Calculus for Dummies 76

Chapter 5. B´ezout’s Theorem 82
1. A Sketch of the Proof 82
2. An Illuminating Example 88

PART II. ELLIPTIC CURVES AND ALGEBRA

Chapter 6. Transition to Elliptic Curves 95

Chapter 7. Abelian Groups 100
1. How Big Is Infinity? 100
2. What Is an Abelian Group? 101
3. Generations 103
4. Torsion 106
5. Pulling Rank 108
Appendix: An Interesting Example of Rank and Torsion 110

Chapter 8. Nonsingular Cubic Equations 116
1. The Group Law 116
2. Transformations 119
3. The Discriminant 121
4. Algebraic Details of the Group Law 122
5. Numerical Examples 125
6. Topology 127
7. Other Important Facts about Elliptic Curves 131
5. Two Numerical Examples 133

Chapter 9. Singular Cubics 135
1. The Singular Point and the Group Law 135
2. The Coordinates of the Singular Point 136
3. Additive Reduction 137
4. Split Multiplicative Reduction 139
5. Nonsplit Multiplicative Reduction 141
6. Counting Points 145
7. Conclusion 146
Appendix A: Changing the Coordinates of the Singular Point 146
Appendix B: Additive Reduction in Detail 147
Appendix C: Split Multiplicative Reduction in Detail 149
Appendix D: Nonsplit Multiplicative Reduction in Detail 150

Chapter 10. Elliptic Curves over Q 152
1. The Basic Structure of the Group 152
2. Torsion Points 153
3. Points of Infinite Order 155
4. Examples 156

PART III. ELLIPTIC CURVES AND ANALYSIS

Chapter 11. Building Functions 161
1. Generating Functions 161
2. Dirichlet Series 167
3. The Riemann Zeta-Function 169
4. Functional Equations 171
5. Euler Products 174
6. Build Your Own Zeta-Function 176

Chapter 12. Analytic Continuation 181
1. A Difference that Makes a Difference 181
2. Taylor Made 185
3. Analytic Functions 187
4. Analytic Continuation 192
5. Zeroes, Poles, and the Leading Coefficient 196

Chapter 13. L-functions 199
1. A Fertile Idea 199
2. The Hasse-Weil Zeta-Function 200
3. The L-Function of a Curve 205
4. The L-Function of an Elliptic Curve 207
5. Other L-Functions 212

Chapter 14. Surprising Properties of L-functions 215
1. Compare and Contrast 215
2. Analytic Continuation 220
3. Functional Equation 221

Chapter 15. The Conjecture of Birch and
Swinnerton-Dyer 225
1. How Big Is Big? 225
2. Influences of the Rank on the Np’s 228
3. How Small Is Zero? 232
4. The BSD Conjecture 236
5. Computational Evidence for BSD 238
6. The Congruent Number Problem 240
Epilogue 245
Retrospect 245
Where DoWe Go from Here? 247

Bibliography 249
Index 251

Read More Show Less

Customer Reviews

Be the first to write a review
( 0 )
Rating Distribution

5 Star

(0)

4 Star

(0)

3 Star

(0)

2 Star

(0)

1 Star

(0)

Your Rating:

Your Name: Create a Pen Name or

Barnes & Noble.com Review Rules

Our reader reviews allow you to share your comments on titles you liked, or didn't, with others. By submitting an online review, you are representing to Barnes & Noble.com that all information contained in your review is original and accurate in all respects, and that the submission of such content by you and the posting of such content by Barnes & Noble.com does not and will not violate the rights of any third party. Please follow the rules below to help ensure that your review can be posted.

Reviews by Our Customers Under the Age of 13

We highly value and respect everyone's opinion concerning the titles we offer. However, we cannot allow persons under the age of 13 to have accounts at BN.com or to post customer reviews. Please see our Terms of Use for more details.

What to exclude from your review:

Please do not write about reviews, commentary, or information posted on the product page. If you see any errors in the information on the product page, please send us an email.

Reviews should not contain any of the following:

  • - HTML tags, profanity, obscenities, vulgarities, or comments that defame anyone
  • - Time-sensitive information such as tour dates, signings, lectures, etc.
  • - Single-word reviews. Other people will read your review to discover why you liked or didn't like the title. Be descriptive.
  • - Comments focusing on the author or that may ruin the ending for others
  • - Phone numbers, addresses, URLs
  • - Pricing and availability information or alternative ordering information
  • - Advertisements or commercial solicitation

Reminder:

  • - By submitting a review, you grant to Barnes & Noble.com and its sublicensees the royalty-free, perpetual, irrevocable right and license to use the review in accordance with the Barnes & Noble.com Terms of Use.
  • - Barnes & Noble.com reserves the right not to post any review -- particularly those that do not follow the terms and conditions of these Rules. Barnes & Noble.com also reserves the right to remove any review at any time without notice.
  • - See Terms of Use for other conditions and disclaimers.
Search for Products You'd Like to Recommend

Recommend other products that relate to your review. Just search for them below and share!

Create a Pen Name

Your Pen Name is your unique identity on BN.com. It will appear on the reviews you write and other website activities. Your Pen Name cannot be edited, changed or deleted once submitted.

 
Your Pen Name can be any combination of alphanumeric characters (plus - and _), and must be at least two characters long.

Continue Anonymously
Sort by: Showing 1 Customer Reviews
  • Posted January 4, 2013

    Excellent Exposition

    Avner and Gross have taught me some new mathematics! They do a good job keeping an abstruse subject matter understandable to folks who are not experts.

    Was this review helpful? Yes  No   Report this review
Sort by: Showing 1 Customer Reviews

If you find inappropriate content, please report it to Barnes & Noble
Why is this product inappropriate?
Comments (optional)