= Mod-three Galois representation of an elliptic curve
{c}
{title2=$\rho_{E,3}:G_K\to\operatorname{GL}_2(\mathbb F_3)$}
Let $E/K$ be an <elliptic curve> with $\operatorname{char}K\ne3$. Its geometric <3-torsion> is a two-dimensional <vector space> over $\mathbb F_3$. Choosing a <basis> gives the continuous <Galois representation>
$$
\rho_{E,3}:G_K\longrightarrow\operatorname{GL}_2(\mathbb F_3).
$$
Here $G_K$ is the <absolute Galois group>, acting on torsion coordinates in a <separable closure>. Its fixed <vectors> are exactly $E(K)[3]$. Thus a nonzero fixed <vector> means a rational <point> of order three; a stable one-dimensional <vector subspace> need only give a Galois-stable cyclic <subgroup>, not a rational generator.
The <Weil pairing> identifies $\bigwedge^2E[3]$ with $\mu_3$, so $\det\rho_{E,3}$ is the mod-three <cyclotomic character>. In particular, the image lies in $\operatorname{SL}_2(\mathbb F_3)$ if $K$ contains all third <roots of unity>.<Elliptic Curves by James S. Milne (2nd edition)>[Chapter V, Weil-pairing determinant argument for torsion representations]{ref}
In <characteristic> three, geometric <3-torsion> is not $\mathbb F_3^2$. The <group scheme> $E[3]$ and its nonreduced structure require a different formulation.
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