| Topology
of Numbers |
The idea is for this to be an introductory textbook on elementary number theory from a geometric point of view, as opposed to the usual strictly algebraic approach. The title "Topology of Numbers" is intended to convey this idea of a more geometric slant, where we are using the word "Topology" in the general sense of "geometrical arrangement" rather than its usual mathematical meaning of a set with certain specified subsets called open sets. A large part of the book is devoted to studying quadratic forms in two variables with integer coefficients, a very classical topic going back to Fermat, Euler, Lagrange, Legendre, and Gauss, but from a geometric viewpoint that emphasizes Conway's much more recent notion of the topograph of a quadratic form. Perhaps the title of the book could have been "Topography of Numbers" but as a topologist I'm partial to "Topology of Numbers".
The current version of the book is still a preliminary draft, so it is incomplete and lacking in polish at certain points.
You can download a pdf file of what currently exists of the book, about 275 pages. This version was posted in July 2019. The main changes from earlier versions occur in the later chapters which have been revised and considerably expanded. There are still a few small topics I would like to add, but apart from this I hope the book is now nearly complete as far as what it covers is concerned. (However, more exercises are needed for the later chapters.)
Chapter 0. Preview
Pythagorean Triples. Rational Points on Other Quadratic Curves. Rational Points on a Sphere. Pythagorean Triples and Quadratic Forms. Pythagorean Triples and Complex Numbers. Diophantine Equations.
Chapter 1. The Farey Diagram
The Diagram. Farey Series. Other Versions of the Diagram. Relation with Pythagorean Triples. The Determinant Rule for Edges.
Chapter 2. Continued Fractions
The Euclidean Algorithm. Connection with the Farey Diagram. The Diophantine Equation ax+by=n. Infinite Continued Fractions.
Chapter 3. Linear Fractional Transformations
Symmetries of the Farey Diagram. Seven Types of Transformations. Specifying Where a Triangle Goes. Continued Fractions Again. Orientations.
Chapter 4. Quadratic Forms
The Topograph. Periodic Separator Lines. Continued Fractions Once More. Pell's Equation.
Chapter 5. Classification of Quadratic Forms
Hyperbolic Forms. Elliptic Forms. Parabolic and 0-Hyperbolic Forms. Equivalence of Forms. Symmetries. The Class Number. Charting All Forms.
Chapter 6. Representations by Quadratic Forms
Three Levels of Complexity. A Criterion for Representability. Representing Primes. Genus and Characters. Representing Non-primes. Proof of Quadratic Reciprocity.
Chapter 7. The Class Group for Quadratic Forms
Multiplication of Forms. The Class Group for Forms. Finite Abelian Groups. Symmetry and the Class Group. Genus and the Class Group.
Chapter 8. Quadratic Fields
Prime Factorization. Unique Factorization via the Euclidean Algorithm. The Correspondences Between Forms and Ideals. The Ideal Class Group. Unique Factorization of Ideals. Applications to Forms.