Integral Points on Elliptic Curves

Abstract

An elliptic curve may have infinitely many rational points, although the Mordell—Weil theorem at least assures us that the group of rational points is finitely generated. Another natural Diophantine question is that of determining, for a given (affine) Weierstrass equation, which rational points actually have integral coordinates. In this chapter we will prove a theorem of Siegel which says that there are only finitely many such integral points. Siegel gave two proofs of his theorem, which we present in sections 3 and 4. Both proofs make use of techniques from the theory of Diophantine approximation, and so do not provide an effective procedure for actually finding all of the integral points. However, his second method of proof reduces the problem to that of solving the so-called “unit equation”, which in turn can be effectively resolved using transcendence theory. We will discuss this method, without giving proofs, in section 5.