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I don't think we need the entire setup below (from Katz and Mazur's elliptic curve book, pages 74 and 75), but, as a beginner, I am unable to identify the assumptions I need.

Let $S$ be the open subset of ${\rm Spec}(\mathbb{Z}[a_1,a_2,a_3,a_4,a_6])$ over which the cubic $y^2+a_1xy+a_3y=x^3+a_xx^2+a_4x+a_6$ is smooth. Let $E$ be the elliptic curve over $S$ given by the equations above.

First, replace $S$ with $S[1/N]$. I'd like to understand why $E[N]$ is etale locally $(\mathbb{Z}/n\mathbb{Z})^2$. At the end of the proof in the book, it says that it suffices to check this at a single geometric point of $S[1/N]$. Why is this true?

So far, I can see that, once you know $[N]:E\rightarrow E$ is etale, we can base change by the identity section $S\rightarrow E$ and get that $E[N]\rightarrow S$ is etale. But then, I don't see why we can go from one geometric fiber (going to be a $\mathbb{C}$-valued point) being $(\mathbb{Z}/n\mathbb{Z})^2$ to all of $E[N]$ being etale locally $(\mathbb{Z}/n\mathbb{Z})^2$.

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  • $\begingroup$ Can't you just explicitly write down an appropriate etale cover by adjoining all of the coordinates of the points in $E[n]$? (Or is there some ramification subtlety here?) $\endgroup$ – Qiaochu Yuan Dec 19 '15 at 3:58
  • $\begingroup$ I've tried to think about this, but I'm not sure how to implement your idea. Is adjoining all the coordinates the same as finding an etale base change that turns $E[n]$ into a disjoint union of schemes? (Can we do this?) $\endgroup$ – Munchlax Dec 19 '15 at 22:27

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