Presentation for a Finite Etale Cover of an (Affine) Elliptic Curve

Question

I posted this question on MSE a few days ago, but I did not get much interest in it. So I thought I would try my luck here. If you are interested in answering the question, there is a bounty over on MSE.

https://math.stackexchange.com/questions/2911318/presentation-for-a-finite-etale-cover-of-an-affine-elliptic-curve

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I will write the question first, then try and explain myself more clearly after:

Question: How can one find a presentation for a finite etale cover of an affine piece of an elliptic curve?

If $E/\mathbb{C}$ is an elliptic curve, then it is homeomorphic to a torus. For this reason we know $\pi_{1}(E) \cong \mathbb{Z}\times \mathbb{Z}$. If we puncture $E/\mathbb{C}$, then we obtain an affine curve, let us denote it $X$. Moreover, $X$ can be thought of as the elliptic curve minus the point at infinity. Since this is affine, we can give a presentation for it, say, $$X:= \text{Spec}(\mathbb{C}[x,y]/\langle y^{2} - f(x) \rangle)$$ for $f(x) = x(x-1)(x-2)$.

We know $\pi_{1}^{et}(X) \cong \hat{\pi}_{1}(X(\mathbb{C})^{an})$ and $X(\mathbb{C})^{an}$ is homeomorphic to a punctured torus. The fundamental group of a punctured torus is the free group on two generators $F_{2}$. The profinite completion of which is non-trivial. So, there are algebraic covers of such an affine curve. How can we get our hands on them? Is there a presentation which in some sense is in terms of $f(x)$? Does anyone have a reference for a discussion on such a construction?

Thanks in advance :)

Answer 1

If you are just looking for examples you can consider the following construction, which is geometric although I don't care to write down the equations explicitly.

Recall, that if $C$ is a compact Riemann surface and $F$ is a reduced divisor of degree $f$ then there is a $f$-fold cyclic cover of $C$, branched at $F$, which we denote by $D$. That is, there is a degree $f$ map: $$ \pi\colon D \rightarrow C $$ which has branching of order $f$ at each of the points of $F$. (Remark: this cover depends on a choice of $f$-th tensor root of the line bundle $\mathcal{O}_C(E)$. There are always $f$-th roots as the map: $$ (-)^{\otimes f}\colon\mathrm{Pic}_1(C) \rightarrow \mathrm{Pic}_f(C) $$ is surjective.)

Using this we can construct the following étale covers of the punctured torus. Start by considering an étale $d$-sheeted cover of your elliptic curve $E$: $$ \phi\colon C \rightarrow E. $$ (Thus, $C$ is a genus 1 curve as well.) Let $x\in E$ be a point and consider a reduced divisor of degree $f$, $F\subset\phi^{-1}(x)$, which is supported on the preimage of $x$ in $C$. Taking the cyclic cover branched at $F$ we get: $$ D\xrightarrow{\pi}C\xrightarrow{\phi}E. $$ Then the composition $\phi\circ \pi$ is a degree $d\cdot f$ map which is étale away from the point $x\in E$.