Tate module ($T_{\ell }(A)={\underleftarrow{\lim }}_{n}A[{\ell }^n]$)

For an abelian variety $A$ and a prime number $\ell$ different from the characteristic, the inverse limit of its geometric ${\ell }^n$-torsion, with transition multiplication by $\ell$, is its Tate module. For an elliptic curve, it is free of rank two over the p-adic integers and satisfies $T_{\ell }(E)/{\ell }^n{T}_{\ell }(E)\simeq E[{\ell }^n]$. The absolute Galois group acts continuously and commutes with all base-defined endomorphisms.

Let $R=K\cap {\mathrm{End}}_k(E)$. If $\alpha \in R$ maps the Tate module into ${\ell }^n{T}_{\ell }(E)$, it kills $E[{\ell }^n]$ and factors through the quotient isogeny $[{\ell }^n]$. The resulting endomorphism is $\alpha /{\ell }^n$, belongs to $K$ and is defined over $k$, hence belongs to $R$. Approximation in $R\otimes {\mathbb{Z}}_{\ell }$ now identifies the displayed stabilizer. This does not assume that the integral Tate module is free over a possibly nonmaximal CM order.

Tensoring a Tate module with the P-adic numbers gives its rational Tate module. For a CM elliptic curve with base-defined multiplication by $K$, it is free of rank one over $K\otimes {\mathbb{Q}}_{\ell }$, including when this algebra is a product of two fields. In the split case the two distinct roots of a nonrational CM endomorphism each have a one-dimensional eigenspace.