A Gateway to Number Theory - Applying the Power of Algebraic Curves

Description

Challenge: Can you find all the integers a, b, c satisfying 2a^2+3b^2=5c^2? Looks simple, and there are in fact a number of easy solutions. But most of them turn out to be anything but obvious! There are infinitely many possibilities, and as any computer will tell you, each of a, b, c will usually be large. So the challenge remains … Find all integers a, b, c satisfying 2a^2+3b^2=5c^2. A major advance in number theory means this book can give an easy answer to this and countless similar questions. The idea behind the approach is transforming a degree-two equation in integer variables a, b, c into a plane curve defined by a polynomial. Working with the curve makes obtaining solutions far easier, and the geometric solutions then get translated back into integers. This method morphs hard problems into routine ones and typically requires no more than high school math. (The complete solution to 2a^2+3b^2=5c^2 is included in the book.) In addition to equations of degree two, the book addresses degree-three equations—a branch of number theory that is today something of a cottage industry, and these problems translate into “elliptic curves”. This important part of the book includes many pictures along with the exposition, making the material meaningful and easy to grasp. This book will fit nicely into an introductory course on number theory. In addition, the many solved examples, illustrations, and exercises make self-studying the book an option for students, thus becoming a natural candidate for a capstone course. Cover Title page Copyright Contents Preface Chapter 1. A Marriage for the Ages 1.1. Mathematics and the Hawaiian Islands 1.2. A Model 1.3. Appearance of the Rational Unit Circle, Part I 1.4. Appearance of the Rational Unit Circle, Part II 1.5. What About Other Rational Circles? 1.6. A Parametric Formula for the Unit Circle in ℚ² 1.7. More General Quadratic Problems 1.8. Conics 1.9. Our Method Also Works in Reverse 1.10. Surveying All Solutions 1.11. The Discriminant 1.12. Finding a Rational Point 1.13. What About Degree 1? Chapter 2. Viewing the Whole Algebraic Curve 2.1. An Ancient Indian Parable 2.2. Viewing Algebraic Curves 2.3. Doing Math at Infinity! 2.4. A More Symmetric Model of ℙ²(ℝ) Chapter 3. Entering the World of Elliptic Curves 3.1. Curves of Degree 3 3.2. What Is an Elliptic Curve? 3.3. Why Irreducible? 3.4. Smoothness 3.5. Weierstrass Forms for Elliptic Curves 3.6. The Discriminant Again 3.7. Our Method Still Works 3.8. Examples of Connecting the Dots 3.9. Mordell’s Theorem Chapter 4. Every Elliptic Curve Is a Group! 4.1. Finite Groups Within an Elliptic Curve 4.2. Finitely Generated Abelian Groups 4.3. Rank 4.4. Mazur’s Theorem 4.5. A Gallery of Positive Rank Elliptic Curves 4.6. How Many Curves? 4.7. Finding Generators Chapter 5. A Million-Dollar Challenge 5.1. Breaking Up a Task into Many Smaller Ones 5.2. The Birch and Swinnerton-Dyer Conjecture 5.3. The Notion of Expected Rank 5.4. Expected Rank of a Random Elliptic Curve 5.5. The Tale of Average Rank 5.6. Rank Results Without GRH or BSD 5.7. About Manjul Bhargava Chapter 6. Every Real Elliptic Curve Lives in a Donut 6.1. Complex Curves 6.2. Complex Numbers Enlighten 6.3. Plotting a Complex Circle 6.4. Plotting a Complex Elliptic Curve 6.5. Subgroups and Cosets 6.6. Elliptic Curves with No Oval Chapter 7. The Genus 7.1. A Few Preliminaries 7.2. Examples 7.3. The Genus Formula 7.4. The Genus vs. Number Theory 7.5. The Curious Story of Plane vs. Fancy Curves Chapter 8. In Conclusion . . . 8.1. Degree 1 8.2. Degree 2 8.3. Degree 3 Appendix A. What Is a Smooth Complex Curve? Appendix B. Algebraic Curves in the Disk Model A Teachable Picture Some Basic Definitions Further Examples Appendix C. Some Code for This Book’s Programs Bibliography Index Back Cover