Ramification divisor with base change

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?

• 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 $B$. – R. van Dobben de Bruyn May 25 at 18:10
• Thank you so much for finding out the problem! – Sheng Meng May 25 at 20:10
• 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! – alpoge May 25 at 23:32
• 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 $y=\infty$. – Sheng Meng May 26 at 11:10