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 $\Bbb Q\_p$)



Possible groups of K-rational points for elliptic curves over arbitrary fields
By Mordell-Weil, for any number field $K$ we have


$$C(K)=\mathbb{Z}^r \times E(K)\_{\mathrm{tors}}$$


As you mention, Mazur showed all the possible options for $E(\mathbb{Q})\_{\mathrm{tors}}$ in his famous 1977 paper.


The only other $K$ for which we have a torsion theorem are the quadratic fields. This is the result of a long series of papers by Kamienny, Kenku and Momose from 1982 to 1992. In particular,


$$\begin{equation}
 E(K)\_{\mathrm{tors}} \cong
 \begin{cases}
 \mathbb{Z}/m\mathbb{Z} & \text{for} 1\leq m\leq 18, m\neq 17 \\
 \mathbb{Z}/2\mathbb{Z} \oplus\mathbb{Z}/2m\mathbb{Z} & \text{for}\,\, 1\leq m\leq 6 \\
\mathbb{Z}/3\mathbb{Z} \oplus\mathbb{Z}/3m\mathbb{Z} & \text{for}\,\, 1\leq m\leq 2 \\
\mathbb{Z}/4\mathbb{Z}\oplus\mathbb{Z}/4\mathbb{Z}
 \end{cases}
 \end{equation}$$


For number fields of degree $>2$ the problem is open, although quite a lot is known for degree $\leq 5$. A (probably not up-to-date) survey on partial results:


* Andrew Sutherland, "Torsion subgroups of elliptic curves over number fields" (2012)


For $K$ a local field, I think it is an old result of Mattuck that $C(K)$ is a topologically finitely generated abelian profinite group. Not sure if anything else is known.


