Number Theory and Geometry An Introduction to 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.

Number Theory and Geometry  An Introduction to Arithmetic Geometry

Number Theory and Geometry An Introduction to Arithmetic Geometry

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.

More Books:

Number Theory and Geometry: An Introduction to Arithmetic Geometry
Language: en
Pages: 488
Authors: Álvaro Lozano-Robledo
Categories: Arithmetical algebraic geometry
Type: BOOK - Published: 2019-03-21 - Publisher: American Mathematical Soc.

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
An Invitation to Arithmetic Geometry
Language: en
Pages: 397
Authors: Dino Lorenzini
Categories: Arithmetical algebraic geometry
Type: BOOK - Published: 1996-02-22 - Publisher: American Mathematical Soc.

Extremely carefully written, masterfully thought out, and skillfully arranged introduction ... to the arithmetic of algebraic curves, on the one hand, and to the algebro-geometric aspects of number theory, on the other hand. ... an excellent guide for beginners in arithmetic geometry, just as an interesting reference and methodical inspiration
Number, Shape, & Symmetry
Language: en
Pages: 444
Authors: Diane L. Herrmann, Paul J. Sally, Jr.
Categories: Mathematics
Type: BOOK - Published: 2012-10-18 - Publisher: CRC Press

Through a careful treatment of number theory and geometry, Number, Shape, & Symmetry: An Introduction to Number Theory, Geometry, and Group Theory helps readers understand serious mathematical ideas and proofs. Classroom-tested, the book draws on the authors’ successful work with undergraduate students at the University of Chicago, seventh to tenth
Homotopy Theory and Arithmetic Geometry – Motivic and Diophantine Aspects
Language: en
Pages: 218
Authors: Frank Neumann, Ambrus Pál
Categories: Mathematics
Type: BOOK - Published: 2021-10-31 - Publisher: Springer

This book provides an introduction to state-of-the-art applications of homotopy theory to arithmetic geometry. The contributions to this volume are based on original lectures by leading researchers at the LMS-CMI Research School on ‘Homotopy Theory and Arithmetic Geometry - Motivic and Diophantine Aspects’ and the Nelder Fellow Lecturer Series, which
An Introduction to the Geometry of Numbers
Language: en
Pages: 345
Authors: J.W.S. Cassels
Categories: Mathematics
Type: BOOK - Published: 1996-12-16 - Publisher: Springer Science & Business Media

From the reviews: "A well-written, very thorough account ... Among the topics are lattices, reduction, Minkowskis Theorem, distance functions, packings, and automorphs; some applications to number theory; excellent bibliographical references." The American Mathematical Monthly