Saturation of Mordell-Weil groups of elliptic curves over number fields
Given a subgroup B of a finitely-generated abelian group A, the saturation B of B is defined to be the largest subgroup of A containing B with finite index. In this thesis we consider a crucial step in the determination of the Mordell-Weil group of an elliptic curve, E(K). Methods such as Descent may produce subgroups H of E(K) with [H:H] > 1. We have determined an algorithm for calculating H given H, and hence for completing the process of finding the Mordell-Weil group. Our method has been implemented in MAGMA with two versions of the programs; one for general number fields K and the other for Q. It builds upon previous work by S. Siksek. Our problem splits into two. First we can use geometry of numbers arguments to establish an upper bound N for the index [H:H]. Second for each remaining prime p < N we seek to prove either that H is p-saturated, i.e. p|[H:H], or to enlarge H by index p. To solve the first problem, 1. We have devised and implemented an algorithm that searches for points on E(K) up to a specified naive height bound. 2. We have devised and implemented an algorithm that calculates the subgroup Egr(K) of points with good reduction at specified valuations. 3. We have implemented joint work with S. Siksek and J. Cremona to calculate an upper bound on the difference of the canonical and naive height of points on an elliptic curve. 4. We have helped to devise and have implemented joint work with S. Siksek and J. Cremona to calculate a lower bound on the canonical heights of non-torsion points on E(K) with K a totally real field. To solve the second problem, 1. As in earlier work by Siksek, we use homomorphisms to prove p-saturation for primes p. We however use the Tate-Lichtenbaum pairing, and we show that, using this pairing, our method will always prove H is p-saturated if that is the case. 2. We show that Siksek's original method will fail for some curves.