Normally, when you define an Elliptic Curve over a field $F$ - it means that the coefficients of the Curve equation lie in field $F$. However, in most cases, I have seen that even the $x$ & $y$ coordinates of the points on the curve also lie in the same field.
I am reading up on Torsion Points from "Mathematical Cryptography" by Jeffrey Hoffstein, Jill Pipher & Joseph H. Silverman & I found that the above isn't true in all cases.
Let $m \ge 1$ be an integer. A point $P \in E$ satisfying $mP = \mathcal{O}$ is called a point of order $m$ in the group $E$. We denote the set of points of order $m$ by $E[m] = \{P \in E : [m]P = \mathcal {O} \}$. Such points are called points of finite order or torsion points. It is easy to see that if $P$ and $Q$ are in $E[m]$, then $P + Q$ and $−P$ are also in $E[m]$, so $E[m]$ is a subgroup of $E$. If we want the coordinates of $P$ to lie in a particular field $K$, for example in $Q$ or $R$ or $C$ or $F_p$, then we write $E(K)[m]$.
From this, it seems as if the group where the coordinates of the points lie may be different from the field over which the curve equation itself is defined.
Why you would want to do this & why would you want to do this for Torsion Points?