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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?

ellCurves
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1 Answers1

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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 $E(K)[m]$.

Álvaro Lozano-Robledo
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