18 704 Seminar In Algebra And Number Theory Rational Points On Elliptic Curves Fall 2004

Elliptic curves lie at the fascinating intersection of geometry, algebra, and cryptography, yet their mathematical beauty can often feel locked behind overly dense academic jargon. This course offers a clear, structured pathway to understanding how to find, analyze, and work with rational points on these curves. By reading through our step-by-step explanations and working through targeted written exercises, you will develop a strong conceptual grasp of the arithmetic of elliptic curves. You will transition from basic algebraic definitions to analyzing complex group structures and understanding modern cryptographic applications. What you'll learn: - Understand the fundamental geometric and algebraic definitions of elliptic curves - Apply the group law to add points on a curve both geometrically and algebraically - Explore the properties of rational points and the significance of Mordell's Theorem - Analyze torsion points and their algebraic structure over different fields - Discover how elliptic curves form the backbone of modern public-key cryptography - Practice calculating points and verifying curve properties through structured written exercises The course begins with foundational algebraic geometry concepts before introducing the chord-and-tangent method for point addition. You will then progress to studying rational points, torsion subgroups, and real-world cryptographic implementations through clear text-based explanations and step-by-step proofs. This course is designed for undergraduate mathematics students, self-taught learners, and computer science enthusiasts with a basic background in algebra. No prior knowledge of advanced algebraic geometry is required. Start exploring the elegant world of elliptic curves and rational points today.