📘 Rational Points on Elliptic Curves (2nd Edition)
Rational Points on Elliptic Curves by Joseph H. Silverman and John T. Tate offers a beautiful blend of algebra, geometry, analysis, and number theory. This textbook serves as both an elegant introduction to elliptic curves and an accessible doorway into arithmetic geometry and Diophantine equations. Written in a friendly and informal style, it invites readers of all levels to appreciate the unity of modern mathematics.
The book is designed for advanced undergraduates and assumes only foundational mathematical knowledge. It is rich in exercises and aims to make advanced topics approachable using tools commonly taught in undergraduate courses.
🎯 What You’ll Learn
- 🧮 The geometry and group structure of elliptic curves
- 🔢 The Nagell–Lutz Theorem for classifying torsion points
- 📐 The Mordell–Weil Theorem on the finite generation of rational points
- 🔍 The Thue–Siegel Theorem on integer points
- 📊 Methods for counting points over finite fields
- 🔐 Lenstra’s Elliptic Curve Factorization Algorithm
- 🧠 A glimpse into Complex Multiplication and Galois Representations
- 🔐 Introduction to Elliptic Curve Cryptography (ECC)
- 🧩 A brief discussion on Fermat’s Last Theorem and its connection to elliptic curves
📦 Structure in This Repository
This section of the repository contains:
courses/: Guided chapter notes and explanationsexamples/: Demonstrations and Rust implementations of algorithmsexercices/: Exercise solutions and walkthroughsimages/: Visual diagrams and mathematical illustrations
🆕 Status
This module is still under active development. Contributions are welcome as we build out more chapters, examples, and implementations!