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


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You can consider rational maps that are defined over $\mathbb Q$, but it's not really correct to talk about an isogeny from $E_1(\mathbb Q)$ to $E_2(\mathbb Q)$, since $E(\mathbb Q)$ is a set of points on the variety $E$, while an isogeny is a type of map between geometric objects. In any case, it is certainly interesting to talk about the field of definition of an isogeny. For example, consider the curve $E:y^2=x^3+x$. The only isogenies $E\to E$ defined over $\mathbb Q$ are the usual multiplication-by-$m$ maps, but if you allowed isogenies defined over $\mathbb Q(i)$, then you get new ones, such as $(x,y)\to(-x,iy)$. So one would write $\text{End}_{\mathbb Q}(E)=\mathbb Z$, but $\text{End}_{\mathbb Q(i)}(E)=\mathbb Z[i]$.


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