Number Theory and Geometry by Alvaro Lozano-Robledo
Number Theory and Geometry - Table of Contents
1. Introduction
PART 1. INTEGERS, POLYNOMIALS, LINES, AND CONGRUENCES
2. The Integers
3. The Prime Numbers
4. Congruences
5. Groups, Rings, and Fields
6. Finite Fields
7. The Theorems of Wilson, Fermat, and Euler
8. Primitive Roots
PART 2. QUADRATIC CONGRUENCES AND QUADRATIC EQUATIONS
9. An Introduction to Quadratic Equations
10. Quadratic Congruences
11. The Hasse–Minkowski Theorem
12. Circles, Ellipses, and the Sum of Two Squares Problem
13. Continued Fractions
14. Hyperbolas and Pell’s Equation
PART 3. CUBIC EQUATIONS AND ELLIPTIC CURVES
15. An Introduction to Cubic Equations
16. Elliptic Curves
What You Will Learn in Number Theory and Geometry
"Number Theory and Geometry: An Introduction to Arithmetic Geometry" by Álvaro Lozano-Robledo explores the rich interplay between "number theory", "geometry", and algebraic structures. The book shows how abstract arithmetic concepts can be understood through geometric interpretations, making complex mathematical ideas more intuitive. It provides a clear introduction to key topics, blending rigorous theory with illustrative examples to guide the reader through advanced concepts.
A central focus of the book is "elliptic curves", "modular forms", and their applications in solving Diophantine equations. Lozano-Robledo explains how these structures connect arithmetic properties with geometric shapes, demonstrating the power of "arithmetic geometry" in uncovering deep patterns in numbers. Each chapter builds upon the last, carefully developing the tools needed to understand the intricate relationship between algebraic techniques and geometric intuition.
Designed for advanced undergraduates, graduate students, and researchers, the book balances theory and application. It emphasizes the geometric perspective in number theory, showing how abstract problems can often be visualized and solved geometrically. Through clear explanations, examples, and exercises, "Number Theory and Geometry" equips readers with the foundations to explore modern research in arithmetic geometry, connecting "number theory", "elliptic curves", and geometric thinking in a coherent, accessible way.
Book Details & Specifications
Title:
Number Theory and Geometry by Alvaro Lozano-Robledo
Publisher:
American Mathematical Society (AMS)
Year:
2019
Pages:
501
Type:
PDF
Language:
English
ISBN-10 #:
147045016X
ISBN-13 #:
978-1470450168
License:
External Educational Resource (with the permission of AMS)
Amazon:
Amazon
About the Author: Alvaro Lozano-robledo
The author
Alvaro Lozano-robledo
is a Spanish-born "mathematician" and professor at the University of Connecticut, specializing in "number theory" and "arithmetic geometry". He has published extensively on elliptic curves, Galois representations, and Diophantine problems, combining deep research with accessible teaching and online resources. In "Number Theory and Geometry", Lozano-Robledo explores the connection between numbers and geometric structures, showing how geometric methods help solve number-theoretic problems. His work bridges rigorous mathematics and practical understanding for students and researchers.
Read or Downloadable Number Theory and Geometry
Free Geometry Books PDF | Curated Academic Library Index