Let $E/\mathbb{Q}$ be an elliptic curve with CM from an imaginary quadratic field $K$. Let $K(E[m])$ denote $m$-th division field (number field obtained by adjoining the coordinates of the $m$-torsion points of $E$. Then if $m=p\_1^{r\_1}\cdots p\_k^{r\_k}$ where $p\_i$, $i=1,2,\cdots,k$ are prime numbers, can we say that $Gal(K(E[m])/K)\cong Gal(K(E[p\_1^{r\_1}])/K)\times\cdots\times Gal(K(E[p\_k^{r\_k}])/K)$?


If yes, then is it easier to directly prove the isomorphism above or the isomorphism
$Gal(K(E[p\_1^{r\_1}])\cdots K(E[p\_k^{r\_k}])/K)\cong Gal(K(E[p\_1^{r\_1}])/K)\times\cdots\times Gal(K(E[p\_k^{r\_k}])/K)$?


I understand that the second isomorphism will only hold if $K(E[p\_i^{r\_i}])\cap K(E[p\_j^{r\_j}])=K$, where $1\leq i,j\leq k$, $i\neq j$ and in fact this actually is the case for some non-CM elliptic curves which I recently found in the book "The decomposition of primes torsion point fields" by Clemens Adelmann. But does it also hold for curves in the CM case?


My intuition:


Since it is true that $K(E[m])\cong K(E[p\_1^{r\_1}])\cdots K(E[p\_k^{r\_k}])$, i.e, it is isomorphic to the compositum of the field extensions $K(E[p\_i^{r\_i}])$, $1\leq i\leq k$, so these extensions $K(E[p\_i^{r\_i}])$ are linearly disjoint from each other and that would also imply that the intersection of any two of these $(i\neq j)$ is $K$?


Is this correct?



Decomposition of the Galois group of the $m$-th division field of an elliptic curve with CM into a direct product of Galois groups
It is not in general true that
$$ {\rm Gal}(K(E[m])/K) \cong \prod\_{i=1}^{k} {\rm Gal}(K(E[p\_{i}^{r\_{i}}])/K)$$
for elliptic curves $E/\mathbb{Q}$ which have CM by an order in $K$. One reason for this is the typical reason: the square root of the discriminant of $E$ is contained in $K(E[2])$, and the square root of the discriminant is also contained in a cyclotomic field, which often (but not always) leads to a non-trivial intersection between $K(E[2])$ and $K(E[m])$ for some odd $m$.


You can find much more about this in the 2022 Pacific Journal of Mathematics paper by Campagna and Pengo (the arXiv version is [here](https://arxiv.org/abs/2006.00883)). In this paper, they show (in Theorem 1.3) that (ignoring the cases of $j = 0$ and $j = 1728$), there are only finitely many elliptic curves $E/\mathbb{Q}$ with CM for which the isomorphism
$$ {\rm Gal}(K(E[m])/K) \cong \prod\_{i=1}^{k} {\rm Gal}(K(E[p\_{i}^{r\_{i}}])/K)$$
holds for all positive integers $m$.


