The translation map between elliptic curves is a rational map

Question

I want to see a reference or a prove that the following map is a rational map:

Let E be an elliptic curve,$P\in E$ and $T_p$ defined as

$T_p:E\rightarrow E,\text{ }T_P(Q)=P+Q$.

It is important that the prove based on the definition of rational maps. See here for example.

Thanks.

Answer 1

Since $+$ is a rational map $E \times E \to E$, and this map is the restriction of $+$ to $E \times \{P\}$, it is a rational map.