Quick question about Chapter 3 Corollary 6.4 [p. 86] in Silverman's Arithmetic of Elliptic Curves. I feel like I'm misreading it and would like clarification.
He claims that for an elliptic curve E defined over K That
$E[m] = \mathbb{Z}/m\mathbb{Z} \times \mathbb{Z}/m\mathbb{Z}$
where $K$ is a perfect field and if it has characteristic $p$, then $p \nmid m$ is also required.
What I don't see is why is it an equality. Couldn't one have a curve like
$E: y^{2} = x(x^{2}+1)$
Then looking modulo $7$ (so considering the previous curve over $\mathbb{F}_{7}$) we have that this equation does not split and so $E[2] \not\cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$? What am I misunderstanding?
By definition, $E[m]$ is the group of $m$-torsion points over the algebraic closure of the field of definition of the curve $E$, i.e., $E[m]$ is what you may also write as $E(\overline{K})[m]$ (the definition of $E[m]$ is in Chapter III, $\S 4$). If instead you want to specify the $m$-torsion subgroup over a non-algebraically closed field, then we do indicate the field and write $E(K)[m]$.