Number Theory and Geometry: An Introduction to Arithmetic Geometry

Item Successfully Added to Cart

An error was encountered while trying to add the item to the cart. Please try again.

Continue Shopping

Go to Checkout

Please make all selections above before adding to cart

Share this page via the icons above, or by copying the link below:

Copy To Clipboard

Successfully Copied!

Number Theory and Geometry: An Introduction to Arithmetic Geometry

Álvaro Lozano-Robledo : University of Connecticut, Storrs, CT

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

List Price:	$99.00
MAA Member Price:	$89.10
AMS Member Price:	$79.20

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

List Price:	$85.00
MAA Member Price:	$76.50
AMS Member Price:	$68.00

Hardcover ISBN:	978-1-4704-5016-8
eBook: ISBN:	978-1-4704-5190-5
Product Code:	AMSTEXT/35.B

List Price:	$184.00 $141.50
MAA Member Price:	$165.60 $127.35
AMS Member Price:	$147.20 $113.20

Add to cart

Click above image for expanded view

Number Theory and Geometry: An Introduction to Arithmetic Geometry

Álvaro Lozano-Robledo : University of Connecticut, Storrs, CT

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

List Price:	$99.00
MAA Member Price:	$89.10
AMS Member Price:	$79.20

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

List Price:	$85.00
MAA Member Price:	$76.50
AMS Member Price:	$68.00

Hardcover ISBN:	978-1-4704-5016-8
eBook ISBN:	978-1-4704-5190-5
Product Code:	AMSTEXT/35.B

List Price:	$184.00 $141.50
MAA Member Price:	$165.60 $127.35
AMS Member Price:	$147.20 $113.20

Add to cart

Book Details
Table of Contents
Additional Material
Reviews
Requests

Pure and Applied Undergraduate Texts

Volume: 35; 2019; 488 pp

MSC: Primary 11; 14

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.

Cover
Title page
Preface
Chapter 1. Introduction
1.1. Roots of Polynomials
1.2. Lines
1.3. Quadratic Equations and Conic Sections
1.4. Cubic Equations and Elliptic Curves
1.5. Curves of Higher Degree
1.6. Diophantine Equations
1.7. Hilbert’s Tenth Problem
1.8. Exercises
Part 1 . Integers, Polynomials, Lines, and Congruences
Chapter 2. The Integers
2.1. The Axioms of \Z
2.2. Consequences of the Axioms
2.3. The Principle of Mathematical Induction
2.4. The Division Theorem
2.5. The Greatest Common Divisor
2.6. Euclid’s Algorithm to Calculate a GCD
2.7. Bezout’s Identity
2.8. Integral and Rational Roots of Polynomials
2.9. Integral and Rational Points in a Line
2.10. The Fundamental Theorem of Arithmetic
2.11. Exercises
Chapter 3. The Prime Numbers
3.1. The Sieve of Eratosthenes
3.2. The Infinitude of the Primes
3.3. Theorems on the Distribution of Primes
3.4. Famous Conjectures about Prime Numbers
3.5. Exercises
Chapter 4. Congruences
4.1. The Definition of Congruence
4.2. Basic Properties of Congruences
4.3. Cancellation Properties of Congruences
4.4. Linear Congruences
4.5. Systems of Linear Congruences
4.6. Applications
4.7. Exercises
Chapter 5. Groups, Rings, and Fields
5.1. \Z/𝑚\Z
5.2. Groups
5.3. Rings
5.4. Fields
5.5. Rings of Polynomials
5.6. Exercises
Chapter 6. Finite Fields
6.1. An Example
6.2. Polynomial Congruences
6.3. Irreducible Polynomials
6.4. Fields with 𝑝ⁿ Elements
6.5. Fields with 𝑝² Elements
6.6. Fields with 𝑠 Elements
6.7. Exercises
Chapter 7. The Theorems of Wilson, Fermat, and Euler
7.1. Wilson’s Theorem
7.2. Fermat’s (Little) Theorem
7.3. Euler’s Theorem
7.4. Euler’s Phi Function
7.5. Applications
7.6. Exercises
Chapter 8. Primitive Roots
8.1. Multiplicative Order
8.2. Primitive Roots
8.3. Universal Exponents
8.4. Existence of Primitive Roots Modulo 𝑝
8.5. Primitive Roots Modulo 𝑝^{𝑘}
8.6. Indices
8.7. Existence of Primitive Roots Modulo 𝑚
8.8. The Structure of (\Z/𝑝^{𝑘}\Z)^{×}
8.9. Applications
8.10. Exercises
Part 2 . Quadratic Congruences and Quadratic Equations
Chapter 9. An Introduction to Quadratic Equations
9.1. Product of Two Lines
9.2. A Classification: Parabolas, Ellipses, and Hyperbolas
9.3. Rational Parametrizations of Conics
9.4. Integral Points on Quadratic Equations
9.5. Exercises
Chapter 10. Quadratic Congruences
10.1. The Quadratic Formula
10.2. Quadratic Residues
10.3. The Legendre Symbol
10.4. The Law of Quadratic Reciprocity
10.5. The Jacobi Symbol
10.6. Cipolla’s Algorithm
10.7. Applications
10.8. Exercises
Chapter 11. The Hasse–Minkowski Theorem
11.1. Quadratic Forms
11.2. The Hasse–Minkowski Theorem
11.3. An Example of Hasse–Minkowski
11.4. Polynomial Congruences for Prime Powers
11.5. The 𝑝-Adic Numbers
11.6. Hensel’s Lemma
11.7. Exercises
Chapter 12. Circles, Ellipses, and the Sum of Two Squares Problem
12.1. Rational and Integral Points on a Circle
12.2. Pythagorean Triples
12.3. Fermat’s Last Theorem for 𝑛=4
12.4. Ellipses
12.5. Quadratic Fields and Norms
12.6. Integral Points on Ellipses
12.7. Primes of the Form 𝑋²+𝐵𝑌²
12.8. Exercises
Chapter 13. Continued Fractions
13.1. Finite Continued Fractions
13.2. Infinite Continued Fractions
13.3. Approximations of Irrational Numbers
13.4. Exercises
Chapter 14. Hyperbolas and Pell’s Equation
14.1. Square Hyperbolas
14.2. Pell’s Equation 𝑥²-𝐵𝑦²=1
14.3. Generalized Pell’s Equations 𝑥²-𝐵𝑦²=𝑁
14.4. Exercises
Part 3 . Cubic Equations and Elliptic Curves
Chapter 15. An Introduction to Cubic Equations
15.1. The Projective Line and Projective Space
15.2. Singular Cubic Curves
15.3. Weierstrass Equations
15.4. Exercises
Chapter 16. Elliptic Curves
16.1. Definition
16.2. Integral Points
16.3. The Group Structure on 𝐸(\Q)
16.4. The Torsion Subgroup
16.5. Elliptic Curves over Finite Fields
16.6. The Rank and the Free Part of 𝐸(\Q)
16.7. Descent and the Weak Mordell–Weil Theorem
16.8. Homogeneous Spaces
16.9. Application: The Elliptic Curve Diffie–Hellman Key Exchange
16.10. Exercises
Bibliography
Index
Back Cover

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

Review Copy – for publishers of book reviews

Desk Copy – for instructors who have adopted an AMS textbook for a course

Examination Copy – for faculty considering an AMS textbook for a course

Permission – for use of book, eBook, or Journal content

Accessibility – to request an alternate format of an AMS title