Algorithmic Aspects of Elliptic Curves

The burgeoning field of computational number theory asks for practical algorithms to compute solutions to arithmetic problems. For example, the Mordell–Weil theorem (VIII.6.7) says that the group of rational points on an elliptic curve is finitely generated, and although we still lack an effective algorithm that is guaranteed to find a set of generators, there are algorithms that often work well in practice. Similarly, Siegel’s theorem (IX.3.2.1) says that an elliptic curve has only finitely many S-integral points, but it took 50 years from Siegel’s proof of finiteness to Baker’s theorem giving an effective bound for the height of the largest solution (IX §5). And Baker’s theorem is only the beginning of the story, since it leads to estimates that, although effective, are not practical without the introduction of significant additional ideas.