**Pure and Applied Undergraduate Texts**

Volume: 35;
2019;
488 pp;
Hardcover

MSC: Primary 11; 14;

**Print ISBN: 978-1-4704-5016-8
Product Code: AMSTEXT/35**

List Price: $109.00

AMS Member Price: $87.20

MAA Member Price: $98.10

**Electronic ISBN: 978-1-4704-5190-5
Product Code: AMSTEXT/35.E**

List Price: $109.00

AMS Member Price: $87.20

MAA Member Price: $98.10

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#### Supplemental Materials

# Number Theory and Geometry: An Introduction to Arithmetic Geometry

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*Álvaro Lozano-Robledo*

Geometry and the theory of numbers are as old
as some of the oldest historical records of humanity. Ever since
antiquity, mathematicians have discovered many beautiful interactions
between the two subjects and recorded them in such classical texts as
Euclid's Elements and Diophantus's
Arithmetica. Nowadays, the field of mathematics that studies
the interactions between number theory and algebraic geometry is known
as arithmetic geometry. This book is an introduction to number theory
and arithmetic geometry, and the goal of the text is to use geometry
as the motivation to prove the main theorems in the book. For example,
the fundamental theorem of arithmetic is a consequence of the tools we
develop in order to find all the integral points on a line in the
plane. Similarly, Gauss's law of quadratic reciprocity and the theory
of continued fractions naturally arise when we attempt to determine
the integral points on a curve in the plane given by a quadratic
polynomial equation. After an introduction to the theory of
diophantine equations, the rest of the book is structured in three
acts that correspond to the study of the integral and rational
solutions of linear, quadratic, and cubic curves, respectively.

This book describes many applications including modern applications
in cryptography; it also presents some recent results in arithmetic
geometry. With many exercises, this book can be used as a text for a
first course in number theory or for a subsequent course on arithmetic
(or diophantine) geometry at the junior-senior level.

#### Readership

Undergraduate and graduate students interested in learning and teaching.

#### Reviews & Endorsements

I think this book would be a great foundation for a course which is more inspiring—and perhaps more challenging—than your standard course on elementary number theory.

-- Abbey Bourdon, MAA Reviews

#### Table of Contents

# Table of Contents

## Number Theory and Geometry: An Introduction to Arithmetic Geometry

- Cover Cover11
- Title page iii4
- Preface xiii14
- Chapter 1. Introduction 118
- Part 1 . Integers, Polynomials, Lines, and Congruences 2744
- Chapter 2. The Integers 2946
- 2.1. The Axioms of \Z 2946
- 2.2. Consequences of the Axioms 3148
- 2.3. The Principle of Mathematical Induction 3350
- 2.4. The Division Theorem 3855
- 2.5. The Greatest Common Divisor 4158
- 2.6. Euclid’s Algorithm to Calculate a GCD 4259
- 2.7. Bezout’s Identity 4360
- 2.8. Integral and Rational Roots of Polynomials 4764
- 2.9. Integral and Rational Points in a Line 4865
- 2.10. The Fundamental Theorem of Arithmetic 5168
- 2.11. Exercises 5572

- Chapter 3. The Prime Numbers 6178
- Chapter 4. Congruences 83100
- Chapter 5. Groups, Rings, and Fields 119136
- Chapter 6. Finite Fields 155172
- Chapter 7. The Theorems of Wilson, Fermat, and Euler 167184
- Chapter 8. Primitive Roots 193210
- 8.1. Multiplicative Order 195212
- 8.2. Primitive Roots 200217
- 8.3. Universal Exponents 203220
- 8.4. Existence of Primitive Roots Modulo 𝑝 205222
- 8.5. Primitive Roots Modulo 𝑝^{𝑘} 210227
- 8.6. Indices 214231
- 8.7. Existence of Primitive Roots Modulo 𝑚 220237
- 8.8. The Structure of (\Z/𝑝^{𝑘}\Z)^{×} 222239
- 8.9. Applications 224241
- 8.10. Exercises 230247

- Part 2 . Quadratic Congruences and Quadratic Equations 235252
- Chapter 9. An Introduction to Quadratic Equations 237254
- Chapter 10. Quadratic Congruences 271288
- Chapter 11. The Hasse–Minkowski Theorem 309326
- Chapter 12. Circles, Ellipses, and the Sum of Two Squares Problem 337354
- Chapter 13. Continued Fractions 361378
- Chapter 14. Hyperbolas and Pell’s Equation 393410

- Part 3 . Cubic Equations and Elliptic Curves 411428
- Chapter 15. An Introduction to Cubic Equations 413430
- Chapter 16. Elliptic Curves 437454
- 16.1. Definition 438455
- 16.2. Integral Points 441458
- 16.3. The Group Structure on 𝐸(\Q) 441458
- 16.4. The Torsion Subgroup 447464
- 16.5. Elliptic Curves over Finite Fields 449466
- 16.6. The Rank and the Free Part of 𝐸(\Q) 455472
- 16.7. Descent and the Weak Mordell–Weil Theorem 459476
- 16.8. Homogeneous Spaces 467484
- 16.9. Application: The Elliptic Curve Diffie–Hellman Key Exchange 471488
- 16.10. Exercises 473490

- Bibliography 479496
- Index 483500

- Back Cover Back Cover1506