Topology of Numbers
This is an introduction to elementary number theory from a geometric point of view, as opposed to the usual strictly algebraic approach. 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 perspective that emphasizes Conway's much more recent notion of the topograph of a quadratic form.
You can download a pdf file of the current version of the book, around 350 pages, posted in September 2022. This will be very close to the version to be published by the American Math Society, probably near the end of 2022. The pdf version will continue to be available here for free downloading after the paper copy has been published.
Chapter 0. Preview
Chapter 1. The Farey Diagram
The Mediant Rule.
Chapter 2. Continued Fractions
Finite Continued Fractions.
Infinite Continued Fractions.
Linear Diophantine Equations
Chapter 3. Symmetries of the Farey Diagram
Linear Fractional Transformations.
Translations and Glide Reflections.
Chapter 4. Quadratic Forms
Chapter 5. Classification of Quadratic Forms
The Four Types of Forms.
Equivalence of Forms.
The Class Number.
Symmetries of Forms.
Charting All Forms.
Chapter 6. Representations by Quadratic Forms
Three Levels of Complexity.
Representations in a Fixed Discriminant.
Genus and Characters.
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 Rational Equivalence.
Chapter 8. Quadratic Fields
Unique Factorization via the Euclidean Algorithm.
The Correspondence Between Forms and Ideals.
The Ideal Class Group.
Unique Factorization of Ideals.
Applications to Forms.