A survey on the group of points arising from elliptic curves with a Weierstrass model over a ring

We survey the known group structures arising from elliptic curves defined by Weierstrass models over commutative rings with unity and satisfying a technical condition. For every considered base ring, the groups that may arise depending on the curve coefficients are recalled. When a complete classification is still out of reach, partial results about the group structure and relevant subgroups are provided. Several examples of elliptic curves over the inspected rings are presented, and open questions regarding the structure of their points are highlighted.