Decomposition of the Galois group of the $m$-th division field of an elliptic curve with CM into a direct product of Galois groups

Question

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?

Answer 1

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). 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 $m$.