Let $E/K$ be an elliptic curve over a number field $K$ and let $E[n]$ denote the full $n$-torsion of $E$, for a positive integer $n$. With $End\_K(E)$ we denote the endomorphisms of $E/K$ which are defined over $K$ and similarly we define $End\_K(E[n])$ to be the endomorphisms of $E[n]$ which are defined over $K$. There is a natural restriction map
$$\phi: End\_K(E) \rightarrow End\_K(E[n]).$$
I am interested in the case that the image of $\phi$ contains $Aut\_K(E[n])$, the set of automorphisms of $E[n]$ which are defined over $K$. My question is the following:



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> Given $E/K$, is there always a lower bound $n\_0$, such that for all $n>n\_0$ we have that the image of $\phi$ contains $Aut\_K(E[n])$?
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Are there any differences between non-CM or CM curves?


Are there any differences between $\mathbb Q$ or a general number field $K$?


If the answer to the above question is yes, are there any results or conjectures concerning a global bound $n\_0(K)$ which is valid for all elliptic curves over a fixed $K$?



When are K-automorphisms of the n-torsion of an elliptic curve E/K liftable to K-endomorphisms of E?
There exists a positive integer $n\_0$ (that depends only on $E$ and $K$) such that the map $\phi$ is surjective if $n$ and $n\_0$ are relatively prime. The same is true not only for elliptic curves but for arbitrary abelian varieties over a finitely generated field. See Inv. Math. 79 (1985), 309-322; arXiv 1301.5594 .


