Determination of modular elliptic curves by Heegner points

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 exp...

Heegner Points and Elliptic Curves of Large Rank

Congruences between Heegner Points and Quadratic Twists of Elliptic Curves

We establish a congruence formula between p-adic logarithms of Heegner points for two elliptic curves with the same mod p Galois representation. As a first application, we use the congruence formula when p = 2 to explicitly construct many quadratic twists of analytic rank zero (resp. one) for a wide class of elliptic curves E. We show that the number of twists of E up to twisting discriminant X...

Higher Heegner points on elliptic curves over function fields

Let E be a modular elliptic curve defined over a rational function field k of odd characteristic. We construct a sequence of Heegner points on E, defined over a Zp -tower of finite extensions of k, and show that these Heegner points generate a group of infinite rank. This is a function field analogue of a result of C. Cornut and V. Vatsal.

Heights of Heegner Points on Shimura Curves

Introduction 2 1. Shimura curves 6 1.1. Modular interpretation 6 1.2. Integral models 13 1.3. Reductions of models 17 1.4. Hecke correspondences 18 1.5. Order R and its level structure 25 2. Heegner points 29 2.1. CM-points 29 2.2. Formal groups 34 2.3. Endomorphisms 37 2.4. Liftings of distinguished points 40 3. Modular forms and L-functions 43 3.1. Modular forms 44 3.2. Newforms on X 49 3.3. ...