The Group of Rational Points on the Unit Circle

Abstract

There have been several attempts to present the basic idea of this marvelous proof to a wider mathematical audience ([2, 7, 81). Another recent paper [31 provides more details. Wiles' proof is based on the theory of elliptic curves, i.e., curves defined by cubic equations. A big part of this theory is devoted to understanding the (points whose coordinates are rational numbers) on these curves. The set of rational points on an elliptic curve has a natural group structure, which will be described very briefly later. It is often very difficult, however, to find all the rational points on an elliptic curve. In this paper we take a much easier and more familiar example-the unit circle-and show how to compute the group structure of its rational points. Next, some applications are given. Finally, for comparison, we give a brief summary of known results on the 'group structure for rational points on elliptic curves.