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Suppose I have an elliptic curve $E/\mathbb{Q}$. What does it mean when one says $E$ has a rational $m$-torsion point over $\mathbb{Q}$? What does this mean for the torsion subgroup, $E_{\mathrm{tor}}(\mathbb{Q})$ and $E[m]$?

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It means that there is a point $P$ on $E$ such that $m\cdot P = O$, and the coordinates of $P$ are in $\mathbb{Q}$. –  Álvaro Lozano-Robledo Sep 15 '12 at 2:57
    
All the (rational)torsion points form a group under group law.We call this group torsion subgroup. –  y zhao Sep 15 '12 at 10:50
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One thing to be careful of here is that the notation $E[m]$ usually means the $m$-torsion points of $E(\overline{\mathbb{Q}})$ and not just the rational $m$-torsion points. –  Matt Sep 15 '12 at 13:31

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