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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?

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    $\begingroup$ The base change $B$ will not be smooth. Your argument only shows that $E_1 \to B$ is birational, i.e. $E_1$ is the normalisation of $B$. $\endgroup$
    – R. van Dobben de Bruyn
    Commented May 25, 2019 at 18:10
  • $\begingroup$ Thank you so much for finding out the problem! $\endgroup$
    – Sheng Meng
    Commented May 25, 2019 at 20:10
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    $\begingroup$ Q4: g(a) is the x-coordinate of 2P for either P with x(P) = a. I believe g(a) [aka x(2P)] is just (f’(a)^2 / 4f(a)) - 2a away from the 2-torsion (aka when f(a)\neq 0), and \infty when f(a) = 0, but googling the multiplication by n law (and the division polynomials) will lead you to some fun stuff. [Btw here I’ve used \PSL_2 to move one of the 4 branch points of \pi to \infty and I’ve written the model y^2 = f(x) with f the monic cubic polynomial with zeroes the 3 non-\infty branch points of \pi.] Hope that helps! $\endgroup$
    – alpoge
    Commented May 25, 2019 at 23:32
  • $\begingroup$ Thank you! You are right. I tried $f(a)=a(a+1)(a-1)$ and the graph is link. One can easily see there are 4 ramification points visible and another two hidden when taking $y=\infty$. $\endgroup$
    – Sheng Meng
    Commented May 26, 2019 at 11:10

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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.

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