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Silverman defines the Group of $\mathfrak a$-torsion points of an elliptic curve $E/\mathbb C$ (with $\mathfrak a$ an ideal in $\mathrm{End}(E)$) in Advanced topics of elliptic curves as $$E[\mathfrak a] = \{P\in E\::\: [\alpha] P = 0 \text{ for all } \alpha \in \mathfrak a\}$$ This seems to be a generalization of $E[m]$ for $m\in \mathbb Z$.

But I don't get it.

Multiplication by $m$ is obviously the $m$-fold addition of $P$ with respect to the group law on $E$. But what would be $[i]P$ for a point on an elliptic curve with CM by $\mathbb Z[i]$?

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  • $\begingroup$ Do you know an elliptic curve with CM by $\mathbf Z[i]$? What is an automorphism of order 4 on that curve? $\endgroup$ – Nefertiti May 31 '16 at 10:56
  • $\begingroup$ By the way, I just notice that since $i$ is a unit in $\mathbf Z[i]$, that might not be the most enlightening example to get an idea of how the construction works. $\endgroup$ – Nefertiti May 31 '16 at 11:00
  • $\begingroup$ So like $y^2=x^3+x$ ... ? $\endgroup$ – Dan May 31 '16 at 11:32
  • $\begingroup$ Right. But in this context it might be easier to think of the curve as $\mathbf C/\Lambda$ for a suitable lattice $\Lambda$. $\endgroup$ – Nefertiti May 31 '16 at 11:51
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As Nefertiti pointed out, since $i$ is a unit in whatever ring it sits in, it’s less useful to ask about $i$-torsion points. But let’s take the curve $E:\,y^2=x^3-x$, which does have an automorphism $[i]$, namely $[i](x,y)=(-x,-iy)$. Then you might ask for $E[1+i]$, and that’s just the set of points $P$ on $E$ for which $P+[i](P)=\Bbb O$

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