= 3-torsion
{title2=$E[3]$}
The 3-torsion of an <elliptic curve> $E$ is the kernel of multiplication by three:
$$
E[3]=\{P:3P=O\},
$$
with the field of points specified. If the characteristic is not three, the geometric group over an algebraic closure is $(\mathbb Z/3\mathbb Z)^2$ and has nine points. The subgroup of points rational over the original field can be smaller.
For $E:y^2=x^3+1$ over $\mathbb Q$, the point $P=(0,1)$ has order three. The tangent slope is $3x^2/(2y)=0$, so the elliptic-curve doubling formulas give $2P=(0,-1)=-P$ and hence $3P=O$. This displays a nonzero rational 3-torsion point rather than only asserting its possible existence.
In characteristic three, the geometric-point count need not be nine. The scheme-theoretic multiplication kernel still carries degree information not captured by counting its ordinary points. Thus the characteristic restriction is essential when identifying geometric 3-torsion with two cyclic groups.
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