2
$\begingroup$

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)

$\endgroup$
3
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.