This is a slight continuation of a previous question of mine.
Given an elliptic curve $E$ over $\mathbb{Q}$ which has complex multiplication.
How would one find each $p$ such that $E$ admits a $p$-isogeny? Further more how would one construct such an isogenous curve?
Example: if one considers the Mordell equations $$
E: y^2=x^3+n.
$$
Then we see that $E$ has complex multiplication because of the endomorphism
\begin{align*}
E& \rightarrow E\\
(x,y) &\mapsto (\zeta x,y)
\end{align*} where $\zeta$ is a third root of unity. Can one find for any particular $n$ which $p$-isogeny arise?
Note: A suggestion that “In case the curve has CM. Then $E[p]$ for any prime of good reduction is fully described by cm theory. As in the above example over its cm field $K=\mathbb{Q}(\sqrt{-3})$ we have that $E[p]$ is a free rank 1 module over $\mathcal{O}_K/p$. Hence isogenies over Q can only occur for additive primes.”
If someone could expand this point and explain why it is true this would also be very helpful.
Best Answer
If a prime $p$ is unramified in the CM field, $K$, then the mod $p$ image of the galois representation is equal to the normaliser of a cartan subgroup, hence not contained in a Borel subgroup (by the classification of maximal subgroups of $GL_2(\mathbb{F}_p)$) and thus $E$ cannot admit a $p$-isogeny (an elliptic curve $E/L$ admits a $p$-isogeny if and only if $\bar{\rho}_{E,p}(G_L)$ is contained in a Borel subgroup).
Now why is this the case that the image is contained in the normaliser of a Cartan subgroup? Note that the CM action $\mathcal{O} \to Aut(E[p])$ (defined over $K$) factors through $\mathcal{O}/p\mathcal{O}$. Then in particular the image of $G_K$ contains (in fact it is equal to) a subgroup of $GL_2(\mathbb{F}_p)$ isomorphic to $\mathcal{O}/p\mathcal{O}$ (these are precisely the Cartan subgroups). But CM does not act on $Aut(E[p])$ over $\mathbb{Q}$, so $\bar{\rho}_{E,p}(G_{\mathbb{Q}})$ contains a Cartan subgroup as an index $2$ subgroup and is thus equal to its normaliser (by the classification of maximal subgroups of $GL_2(\mathbb{F}_p)$).
Thus $E$ may only admit a $p$-isogeny if $p$ is ramified in the CM field.