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?



Ramification divisor with base change
So the answer is:


Question 1 is wrong in general as R. van Dobben de Bruyn pointed out in the comment that the base change may not be normal.
So there is no conflict in Main Question 3.


See also the comment by alpoge for Question 4.


