TITLE: Possible groups of K-rational points for elliptic curves over arbitrary fields
QUESTION [10 upvotes]: It is known that the group $C(\Bbb R)$ has at most two connected components, and the connected component of the identity is isomorphic to $U(1)$ as a topological group (trivially) and $C(\Bbb Q)$ is isomorphic to $\Bbb Z^r\times E(\Bbb Q)$, where all possible $E(\Bbb Q)$ are known.
Are there similar results for other fields? I am especially interested in the case K is a number field, or the p-adic numbers (both $\Bbb Z_p$ and $\Bbb Q_p$)

REPLY [16 votes]: The answer to one possible interpretation of the title question -- vary over all elliptic curves over all fields and ask which groups arise -- is given in this paper.
With regard to the structure of Mordell-Weil groups of elliptic curves over local fields, I think you will find that $\S$5.1 of this joint paper with Allan Lacy relevant.  (The title is "Mordell-Weil Groups of Abelian Varieties Over Local Fields".)  The case of characteristic $0$ is a slightly more general and less precise variant of Joe Silverman's answer, but things work a bit differently in positive characteristic.