Let's work over $\mathbb{C}$. Consider the following commutative diagram

\begin{array}{llllllllllll} E_1& \xrightarrow{f} &E_2\\ \downarrow{\pi} &&\downarrow{\pi}\\ P_1 & \xrightarrow{g} &P_2 \end{array} Here $E_1=E_2=E$ is an elliptic curve, $P_1=P_2=\mathbb{P}^1$, $f$ is the multiplication map defined by $f(x)=2x$. Let $\sigma:E \to E$ denote the involution: $\sigma(x)=-x$ and let $\pi:E\to E/<\sigma>=\mathbb{P}^1$ denote the quotient of the involution. Since $\sigma\circ f=f\circ \sigma$ we have the induced endomorphism $g$ to make the diagram commutes.

With the diagram above, I have the following questions:

**Question 1 (Base change).** $E_1=E_2\times_{P_2}P_1$?

My reason: Let $B:=E_2\times_{P_2}P_1$. Then $B\to P_1$ is a (flat) double cover. So $B$ has at most 2 irreducible components. Note that $B\to E_2$ is a finite cover. So each component of $B$ is not rational and hence elliptic and further $B$ is an irreducible elliptic curve. Note that $E_1$ factors through $B$ and $\pi$ is a double cover. So the induced $E_1\to B$ is isomorphic.

**Fact (Pullback of relative sheaf of differential).**

A) $\Omega_{E_1/E_2}=0$ (unramified)

B) $\Omega_{P_1/P_2}\neq 0$ (ramified)

C) $\Omega_{E_1/E_2}=\pi^*\Omega_{P_1/P_2}$

See GTM52, Hartshorne, Chapter II, Proposition 8.10.

**Question 2 (Faithfully flat)** $\pi^*\mathcal{F}=0\Rightarrow \mathcal{F}=0$ for any coherent sheaf $\mathcal{F}$ of $P_1$?

My reason: $\pi$ is flat and surjective and hence faithfully flat.

**Main Question 3(Contradiction).** Question 2 and Facts A) and C) imply $\Omega_{P_1/P_2}=0$ which contradicts Fact B)? Where am I wrong?

**Question 4.** What is the rational function of $g^*(x)$ where we assume $\mathbb{C}(x)$ is the function field of $\mathbb{P}^1$?

I know it depends on $\pi$, but is $\frac{x^2}{(x+1)^2(x-1)^2}$ possible?