Franz Lemmermeyer

Pell Conics

An Alternative Approach to Elementary Number Theory

In this book I would like to present the content of elementary number theory using an approach based on Pell conics. The idea is to prepare the readers for more advanced material on elliptic curves or algebraic geometry at a very early stage. Comments and corrections are highly welcome.
  1. Preface (Nov. 2012)
  2. Starting with the Unit Circle (Nov. 2012)
  3. Examples of Conics (Nov. 2012)
  4. Residue Class Rings (Nov. 2012)
  5. Pell Conics over Residue Class Rings
  6. Algorithmic Number Theory
  7. Cryptography
  8. Modularity (Nov. 2012)
  9. Hecke operators and the Herglotz trick
  10. L-Series
  11. The Rank of Pell Conics
  12. The Solvability of the Pell Equation
  13. Modular Pell varieties
  14. References (Nov. 2012)
The second part of this book will discuss the geometric interpretation of the group law, Pascal's Theorem, group laws on degenerate conics and singular cubics, Newtons' method and formal group laws, the group law via the theory of divisors in function fields, and a few other topics.