I am currently working through this paper by Denef, and I have run into a little bit of a snafu with Lemma 3.1.

The setup is as follows: let $K$ be any field of characteristic zero, and let $E_0$ be the elliptic curve defined by the equation

$$Y^2 = X^3 + aX + b$$

Identify $T$ with the rational function $(x,y) \to x$ on $E_0$ and identify $U$ with the rational function $(x,y) \to y$ on $E_0$. Consider the field of rational functions $F = K(T,U)$, where $U^2 = T^3 + aT + b$. Denote the space of $K$-rational maps from $E_0$ to $E_0$ by $Rat_K(E_0,E_0)$ and let $\psi_2$ be the map

$$\psi_2: E_0(F) \to Rat_K(E_0,E_0)$$

which sends the point $(V,W)$ on $E_0(F)$ (the set of rational points of the elliptic curve $Y^2 = X^3 + aX + b$ where, in this case, $(x,y)\in F$) to the $K$-rational map

$$\psi_2(V,W): E_0(K) \to E_0(K): (x,y) \mapsto (V(x,y), W(x,y))$$

The claim is then that $\psi_2$ is a homomorphism. This seems clear to me since if we were to try and construct a rational map on $E_0$, this amount to sending a rational point $(X,Y)$ satisfying

$$Y^2 = X^3 + aX + b$$

to some other point satisfying the same elliptic curve equation, and we know that

$$W(x,y)^2 = V(x,y)^3 + aV(x,y) + b$$

since we made the assumption that these rational maps were members of $F$ (well, technically, they are identified with members of $F$ which are rational functions of $T$ and $U$, but I think that the point is clear).

My question is then how one would explicitly write down the homomorphism described by $\psi_2$? The first place that I am having trouble is trying to figure out how to write down the image of the identity of the elliptic curve would map to the identity map? I tried to write this down explicilty in terms of projective coordinates so that $[T:U:Z] = [0:1:0] \mapsto Id_{E_0}$, but I think that I am either missing or misunderstanding something since this is seeming to be a lot more trouble than I feel like it should be. Any help is greatly appreciated; thank you!



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