Galois module structure of unit groups and Mordell-Weil groups

The Galois group of a Galois extension F/K of number fields acts on the ring of integers of F and on the unit group. If one fixes the isomorphism class of the Galois group, then only finitely many lattices can be realised as these modules. Similarly, if one has an elliptic curve over K such that E(F) is a specific rational representation when tensored with Q, then E(F)/torsion is isomorphic to one of only finitely many possible lattices. This project will investigate, either in the unit group or in the elliptic curve setting or both, which lattices appear with what frequencies, as F varies.