Let $K$ be a field and $E$ be an elliptic curve defined over $K$. It well understood the $K$-points on $E$ forms an abelian group. What is the structure of this group?(Depending on char($K$)?) Is it a direct sum of some well known abelian groups such as $\mathbb{Z}/m\mathbb{Z}$?



Elliptic curves — general structure of the group
First case: Complex numbers. Over $\mathbb C$ the structure as an abstract group is $\mathbb S^1 \oplus \mathbb S^1$ where $\mathbb S^1$ is the circle, i.e., $\mathbb R/\mathbb Z$. This follows as Robin Chapman notes [below](https://mathoverflow.net/questions/33191/elliptic-curves-general-structure-of-the-group#comment75333_33194), i.e., it is a complex torus in the form $\mathbb C/\Lambda$ where $\Lambda$ is a lattice in $\mathbb C$.


Let $K$ be an algebraically closed field of char $p$. If $n$ is prime to $p$, the the $n$-torsion is $\mathbb{Z}/n\mathbb{Z} \oplus \mathbb{Z}/n\mathbb{Z}$. The $p^e$-torsion could be either trivial for all $e$, or $\mathbb{Z}/p^e\mathbb{Z}$ for all $e$.


Over a non-algebraically closed-field, this is going to be much more complicated. I will try to give just an introduction.


Over $\mathbb Q$ and number fields: Over $\mathbb Q$ or a number field, it is finitely generated by the Mordell–Weil theorem. So it has a torsion part and free part. The free part could be arbitrarily large.


The Mordell–Weil theorem is in fact true for arbitrary finitely generated fields. This is due to Néron.


Over $\mathbb Q$, the torsion part has exactly $15$ possibilities according to the [theorem of Mazur](https://planetmath.org/MazursTheoremOnTorsionOfEllipticCurves). Over number fields, this had been generalized that the torsion part is uniformly bounded.


Over finite fields, the torsion group would be $\mathbb{Z}/n\mathbb{Z} \oplus \mathbb{Z}/m\mathbb{Z}$ where $n \mid m$. There is no free part.


And it could go on like this. Please have a look at Silverman's "[Advanced topics in the Arithmetic of Elliptic Curves](https://doi.org/10.1007/978-1-4612-0851-8)" for elliptic curves over real numbers, $p$-adic numbers, function fields, etc.


