Heegner point computations

Abstract

We discuss the computational application of Heegner points to the study of elliptic curves over Q, concentrating on the curves ED: Dy2 = x3 − x arising in the “congruent number” problem. We begin by briefly reviewing the cyclotomic construction of units in real quadratic number fields, which is analogous in many ways to the Heegner-point approach to the arithmetic of elliptic curves, and allows us to introduce some of the key ideas in a more familiar setting. We then quote the key results and conjectures that we shall need concerning elliptic curves and Heegner points, and show how they yield practical algorithms for finding rational points on Ed and other properties of such curves. We conclude with a report on more recent work along similar lines on the elliptic curves x3 + y3 = A.