I was reading Silverman and Tate's Rational Points on Elliptical Curves, and it said something along the lines of
Birational transformation preserves the structure of the groups of the points of rational curves. This follows from the properties of algebraic curves.
What is the proof of the above statement of the invariance of the group under birational transformations? Is there an intuitive approach for understand it?
The intuition behind this rests on two facts:
A morphism $\phi:E_1\to E_2$ between elliptic curves (that sends $O_E$ to $0_{E'}$) sends collinear points to collinear points.
The addition on an elliptic curve is defined so that if $P,Q,R$ are the three intersection points of a line with $E$, then $P+Q+R=0$.
Thus, if $\phi:E_1\to E_2$ is a morphism as above, and $\phi(P)=P'$, $\phi(Q)=Q'$, and $\phi(R)=R'$, it follows that $P'$, $Q'$, and $R'$ are also collinear, and therefore $P'+Q'+R'=0_{E'}$. Hence $$\phi(P)+\phi(Q)+\phi(-(P+Q))=\phi(0_E)$$ or equivalently, $\phi(P)+\phi(Q)=-\phi(-(P+Q))=\phi(P+Q)$, i.e., the morphism is actually a homomorphism of groups. If in addition $\phi$ has an inverse morphism, then we have an isomorphism of groups.
For a formal proof of all this, see Silverman's "The Arithmetic of Elliptic Curves", Chapter III, and in particular Theorem 4.8.