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Suppose that $k$ is an algebraically closed field of characteristic $p > 0$ and $E/k$ is a supersingular elliptic curve equipped with a full level $N$ structure $\phi$ for some $N \ge 3$ that is prime to $p$. Let $(E^{(p^{2n})}, \phi^{(p^{2n})})$ be the $p^{2n}$-Frobenius pullback of $(E, \phi)$. Suppose that one has a $k$-isomorphism $A\colon (E, \phi) \xrightarrow{\sim} (E^{(p^{2n})}, \phi^{(p^{2n})})$. How does it follow "by Galois descent" that $(E, \phi)$ must be defined over $\mathbb{F}_{p^{2n}}$?

This is claimed on top of p. 96 of Katz-Mazur, but I cannot understand what descent theorem they are trying to apply: the extension $k/\mathbb{F}_{p^2}$ is not Galois in general, so it seems that fpqc descent should be involved, but then how does $A$ give rise to a descent datum? I guess that the only role of $\phi$ is to make some cocycle condition automatic due to rigidity of level $N$ structures.

I know that the proof I am referring to may be alternatively carried out by using $j$-invariants, but I would like to understand the reasoning intended by Katz and Mazur, since such a descent technique seems useful to know.

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  • $\begingroup$ I $k$ is an algebraic closure of $\Bbb F_{p^2}$, then it certainly is Galois (though infinite) over the smaller field. $\endgroup$
    – Lubin
    Feb 8, 2015 at 18:52
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    $\begingroup$ Sometimes books contain harmless typos; this is one such. It is almost certain that Katz & Mazur assume the reader knows all supersingular elliptic curves over such $k$ descend to $\overline{\mathbf{F}}_p$ by $j$-invariant reasons, and they invoke it. Don't read too much into what they write (in particular, one should never speak of "Galois descent" for a non-Galois extension). But the rigidity they mention makes descent work verbatim if one says the "fpqc" in place of "Galois" (though it is heavy to do that for this purpose); what is your difficulty to carry that out? $\endgroup$
    – user74230
    Feb 8, 2015 at 18:59
  • $\begingroup$ @user74230: My difficulty is that to carry out fpqc descent I need an isomorphism between the two pullbacks of $(E, \phi)$ along the projections $\mathrm{Spec}(k \otimes_{\mathbb{F}_{p^{2n}}} k) \rightarrow \mathrm{Spec} (k)$ and I don't understand how $A$ is of help for this. For all that I can tell, $k \otimes_{\mathbb{F}_{p^{2n}}} k$ is some huge horrendous ring that I don't understand anything about. $\endgroup$
    – Lisa S.
    Feb 9, 2015 at 5:59
  • $\begingroup$ @LisaS.: Good point, I mistakenly thought (not very carefully...) that rigidity made everything work out, but of course that is nonsense unless one has some maps to work with. So I revert back to my initial comment, namely that it seems clear Katz & Mazur had in mind to easily make descent to some huge but finite field, and then Galois descent works as they say. There actually are situations where the rigidity method makes fpqc descent sail through, but this isn't one of them. $\endgroup$
    – user74230
    Feb 9, 2015 at 7:01

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