Birational transformation of Elliptic curves?

Question

Let $F:V\to W$ be a birational transformation of elliptic curves; let $g$ be a generator of $V$. Is necessarily $F(g)$, a generator of $W$?

Answer 1

No, this is not necessarily the case. Here is a silly example. Using the same argument one can come up with more exotic examples.

An elliptic curve is a pair $(E,\mathcal{O})$ consisting of a smooth projective curve $E$ of genus $1$, and a point $\mathcal{O}$ that will serve as identity of the group. Suppose $V=(E,[0:1:0])$, where $E: y^2=x^3-2$, with $\mathcal{O}=[0:1:0]$ the point at infinity. The group $V(\mathbb{Q})\cong \mathbb{Z}$, generated by $P=(3,5)=[3:5:1]$. Now let $W=(E,P)$, and $F:V\to W$ to be the identity map, i.e., $F(x,y)=(x,y)$. Then, $F(P)=P$, but $P\in V$ has infinite order and generates $V(\mathbb{Q})$, while $P\in W$ has order $1$, being the identity element of $W$, so $F(P)$ does not generate $W(\mathbb{Q})$.