Let $E_1$ and $E_2$ be elliptic curves and $T_l(E_i)\cong \mathbb{Z}_l \oplus \mathbb{Z}_l$ the $l$-adic Tate module.

Given $ \varphi \in Hom(E_1,E_2)$ this induces $\varphi_l \in Hom(T_l(E_1),T_l(E_2))$ and the map $\varphi \mapsto \varphi_l$ is injective.

Why is the natural map $ Hom(E_1,E_2)\otimes \mathbb{Z}_l \longrightarrow Hom(T_l(E_1),T_l(E_2))$ being injective a stronger statement?

(This appears in Silverman's Arithmetic of elliptic curves p89)


1 Answer 1


You have two injections
$\hom(E_1,E_2) \longrightarrow \hom(E_1,E_2) \otimes_{\mathbb Z} \mathbb Z_l \longrightarrow \hom(T_l(E_1),T_l(E_2)) $

Since $\hom(E_1,E_2) \otimes \mathbb Z_l$ is strictly bigger than the image of $\hom(E_1,E_2)$, it is better to know that each map is injective than to know that their composition is injective.


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