Constructing Rational Points on Elliptic Curves Using Heegner Points

These are the notes I wrote for my candidacy talk. The aims for this talk were to understand Heegner points, examine the different ways they can be characterized, and get an idea of how to construct rational points on an elliptic curve using Heegner points. I cite some good references at the end if you are also trying to begin learning about this beautiful topic. The goal of this talk is to explain how Heegner points provide us with a (nontorsion) rational point on a given (rank 1) rational elliptic curve. We will do this using a rather abstract approach, and if time permits, we will explain a more concrete construction which is good for numerical computations. Let Y0(N) be the open modular curve over Q which is defined by, Y0(N)(C) = { (E,E′, φ) : φ : E→ E′ isogeny with kerφ cyclic of orderN }/ ∼, (i.e. its points correspond to N -isogenous pairs of elliptic curves defined over C), (where (E,E′, φ) ∼ (Ẽ, Ẽ′, φ̃) if there exist isomorphisms ψ, ψ′