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Torsion Points under SL_2(Z/nZ)

I would like to rephrase a question I asked three days ago and was closed for the unclear presentation.

Taking $T \in E[p]$, a point in the p-torsion of the elliptic curve and looking at the right action of $SL_2({\mathbb Z}/p{\mathbb Z})$ on $T$ for prime $p$, one can easily see that you obtain again all points in $E[p]$ equally many times. The action is described as follows:

$$ \Big(\frac{m}p \quad \tau+ \frac{n}p \Big) \left(\begin{matrix} a & b \\\ c & d \end{matrix} \right) = \Big(\frac{ma+nc}p \quad \tau+ \frac{mb+nd}p \Big) $$

Now what would happen when we take $T \in E[n]$ where $n$ is an integer and look at the right action of $SL_2({\mathbb Z}/n{\mathbb Z})$? Do we also get again all the points in $E[n]$ equally many times? If not, why?

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  • $\begingroup$ I believe that the right thing to do in such cases is to go to meta, present the new version of the question and ask people to reopen the old question with the revised statement. $\endgroup$
    – José Figueroa-O'Farrill
    Commented May 21, 2011 at 12:38
  • $\begingroup$ Closed question is: mathoverflow.net/questions/65271/… $\endgroup$
    – José Figueroa-O'Farrill
    Commented May 21, 2011 at 12:40
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    $\begingroup$ What is the role of the elliptic curve? I don't think the answer will change if you simply consider the action of $SL_2(\mathbb{Z}/n)$ on $(\mathbb{Z}/n)^2$. $\endgroup$
    – Felipe Voloch
    Commented May 21, 2011 at 12:55
  • $\begingroup$ Do you want $SL_2(\mathbb{Z}/n)$ acting, or $SL_2(\mathbb{Z}/p)$? $\endgroup$
    – David E Speyer
    Commented May 21, 2011 at 13:01
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    $\begingroup$ The action of $SL_2(\mathbb{Z}/n)$ preserves the order of a point, so the statement cannot be true if $n$ is composite. $\endgroup$
    – François Brunault
    Commented May 21, 2011 at 14:54

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Actually I need to know both. First for the action under $SL_2(\mathbb{Z}/p\mathbb{Z})$ and $SL_2(\mathbb{Z}/n\mathbb{Z})$

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    $\begingroup$ This is not an answer, but an (unsuccessful) attempt to clarify the question. You might want to delete this "answer" and edit your question to clarify. The mathematical issue is this: in your question you seem to be saying that $SL_2(\mathbb{Z} / p \mathbb{Z})$ is acting on $E[n]$ -- presumably you mean the action of $SL_2(\mathbb{Z} / p \mathbb{Z})$ on $E[p]$ for $p$ prime, and of $SL_2(\mathbb{Z} / n \mathbb{Z})$ on $E[n]$ for general $n$. $\endgroup$
    – David Loeffler
    Commented May 21, 2011 at 14:39
  • $\begingroup$ See my comment below the question. $\endgroup$
    – S. Carnahan ♦
    Commented May 23, 2011 at 5:39

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