I recently try to read Vatsal's paper ``Multiplicative subgroups of $J_0(N)$ and applications to elliptic curves.'' He seemed to use the following fact freely:

Let $E$ be a semistable elliptic curve defined over $\mathbb{Q}$, and suppose that $E$ admits a cyclic $\ell$-isogeny $\phi : E \to E'$. Then $\phi$ is étale if and only if the kernel of $\phi$ is isomorphic to $\mathbb{Z}/\ell\mathbb{Z}$ as an $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$-module.

Because of the lack of my knowledge about étale isogenies, I cannot figure out the proof of the above fact. In fact, I'm not sure that the statement is even true. If anyone let me know the proof of the above statement, it's very helpful for me to understand Vatsal's paper.

Moreover, I failed to find reasonable references about étale isogenies. Any suggestions about good references will be appreciated.

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    $\begingroup$ He must be using some non-standard definition of etale. $\endgroup$ – Felipe Voloch Nov 4 '12 at 13:47
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    $\begingroup$ "We say that φ is étale if the extension [...] to Néron models is étale." $\endgroup$ – RP_ Nov 4 '12 at 13:55
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    $\begingroup$ Proof of "only if": It suffices that ker($\phi$) is unramified at all primes, so we may work over the fraction field $K$ of the completed max. unramified extension $W$ of each $\mathbf{Z}_p$. For good reduction $N(E_K)$ and $N(E'_K)$ are elliptic curves, so the etale ker($N(\phi_K)$) is finite, thus ker($\phi_K$) unramified (like $K$-fiber of any finite etale $W$-scheme). For mult. reduction, Tate handles $\ell\ne p$, so assume $\ell=p$. Now $N(E_K)[p]$ contains $\mu = \mu_p$ and $\phi_K(\mu_K) \ne 0$ (or else $\mu\subset\ker(N(\phi_K))$, violating $W$-etaleness), so again we win by Tate. $\endgroup$ – user27056 Nov 4 '12 at 16:42
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    $\begingroup$ Proof of "if": False if $\ell = 2$. Assume $\ell > 2$. May again work over $K$. Assume good reduction, so $\ker(N(\phi_K))$ is finite flat of order $\ell$ and $N(\phi_K)$ is etale iff its kernel is etale. The case $\ell \ne p$ is clear, and $\ell = p$ follows by Raynaud's theorem (since $\ell > 2$). Assume mult. reduction. By translations, it's equivalent to show the isogeny between formal groups over $W$ is etale. That is an endomorphism of the formal mult. group of degree 1 or $\ell$, and degree $\ell = p$ puts $\mu_p$ inside ker($\phi_K$), contradicting constancy (as $\ell >2$). QED $\endgroup$ – user27056 Nov 4 '12 at 17:09
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    $\begingroup$ @Will: Choose a semistable $E$ with split multiplicative reduction at 2 and split 2-torsion over $\mathbf{Q}$. Check (by descent from $\mathbf{Q}_2$) that for a suitable point $P$ of order 2, the schematic closure of $\langle P \rangle$ in $N(E)_{\mathbf{Z}_2} = N(E_{\mathbf{Q}_2})$ is $\mu_2$, so the isogeny $E \rightarrow E' := E/\langle P \rangle$ is a counterexample (since Grothendieck's inertial criterion ensures that $E'$ is semistable at all primes). $\endgroup$ – user27056 Nov 4 '12 at 18:00

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