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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:

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 $Aut_K(E[n])$?

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 $K$?

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    $\begingroup$ Suppose in addition that $n = p$ is prime and $E$ does not have CM. Then, for large $p$, Serre's surjectivity theorem implies that $Aut_K(E[p]) = \mathbf{F}_p^*$, and consequently the answer to (this modification of) your question is 'yes'. $\endgroup$
    – Kestutis Cesnavicius
    Commented Nov 6, 2013 at 16:15
  • $\begingroup$ Hey Kestutis, you are of course right. I should have emphasised, that I am especially interested in the CM case. $\endgroup$
    – Stefan Keil
    Commented Nov 6, 2013 at 16:39

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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 .

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    $\begingroup$ Dear Yuri, thanks a lot for your answer and your reference. Remark 5.4.7 in your Inv. Math. paper is exactely what I was looking for. I used the following version: Fix an abelian variety A/K. Then there is a finite set S=S(A,K) of rational primes, such that for all primes powers p^k, with p is not in S, every Galois equivariant endomorphism A[p^k]->A[p^k] is induced by an endomorphism of A. I'd like to reask my third question. Is it conjectured that there is a 'global' finite set S=(d,K) of rational primes, with d a positive integer, such that the above holds for all A/K, such that dim(A)=d? $\endgroup$
    – Stefan Keil
    Commented Nov 8, 2013 at 10:05
  • $\begingroup$ Dear Stefan, you are welcome. As far as I know, nobody stated it as a conjecture for $d>1$. $\endgroup$
    – Yuri Zarhin
    Commented Nov 8, 2013 at 11:10
  • $\begingroup$ And is there a reference where it is conjectured for $d=1$? $\endgroup$
    – Stefan Keil
    Commented Nov 8, 2013 at 12:24
  • $\begingroup$ It seems to me that for elliptic curves the question (for non-CM) elliptic curves $E$ about the Galois image in $Aut(E[p])$ was raised by Serre in his (already mentioned) 1972 Inventiones paper. Some partial (but very important) results in this direction were obtained recently by Yuri Bilu and Pierre Parent annals.math.princeton.edu/2011/173-1/p13 . $\endgroup$
    – Yuri Zarhin
    Commented Nov 17, 2013 at 10:53

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