Mod-three Galois representation of an elliptic curve (${\rho }_{E,3}:G_K\to {\mathrm{GL}}_2({\mathbb{F}}_3)$)

Let $E/K$ be an elliptic curve with $\mathrm{char}K\ne 3$. Its geometric 3-torsion is a two-dimensional vector space over ${\mathbb{F}}_3$. Choosing a basis gives the continuous Galois representationHere $G_K$ 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 ${\bigwedge }^{2}E[3]$ with ${\mu }_3$, so $\det {\rho }_{E,3}$ is the mod-three cyclotomic character. In particular, the image lies in ${\mathrm{SL}}_2({\mathbb{F}}_3)$ if $K$ contains all third roots of unity.^[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.