I am studying Silverman's "The Arithmetic of Elliptic Curves" and I got the following question:

In the first chapters he defines rational between projective varieties (see the first definition in I.3). This maps are actually defined over an algebraic closed field in the sense that a rational map

$$\phi:V_1\rightarrow V_2,\phi=[f_0,\dots f_n]$$ where $f_i\in \overline{K}(V_1)$. Afterwards he gives the definition of rational maps between arbitrary fields i.e. we have then $f_i\in K(V_1)$.

My question is: Does the whole theory (especially about isogenies in Chapter 3) only work for $\overline{K}$? Are we free to take $K=\mathbb Q$ and consider rational maps between $E_1(\mathbb Q)$ and $E_2(\mathbb Q)$?

Thanks in advance