I have a question in X.3 proposition 3.2 of Silverman's book AEC.
Let $E/K$ be an elliptic curve and $C/K$ be a homogeneous space for $E/K$. Fix a point $p_0\in C$ and define a map $\theta: E\rightarrow C$ given by $\theta(P)=p_0+P$, where $+$ means the right action of $E$ over $C$. Then I want to understand why $\theta$ is a morphism defined over $K(p_0)$.
I think a morphism is defined over $K(p_0)$ by definition is invariant under the action of $\text{Gal}(\bar{K}/K(p_0))$. But Silverman proves that $\theta(P)^\sigma=\theta(P^\sigma)$ for all $\sigma\in\text{Gal}(\bar{K}/K)$, I can't see why this imply $\theta$ is defined over $K(p_0)$. Thanks.