Timeline for Kernel of an \'etale isogeny of prime degree $\ell$ between elliptic curves
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15 events
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| Nov 5, 2012 at 22:27 | comment | added | user27056 | @Tim Dokchister: If you look at Vatsal's paper you'll find that he assumes $\ell > 2$, and he records the result you mention as his Lemma 5.2 with proof punted to Serre's paper. I doubt this is what he has in mind for the OP's question, since it is far from a statement about etaleness for a map of Neron models. Vatsal's Lemma 2.2 and Proposition 2.3 essentially include the result about which the OP asks (in other terminology) -- the proofs of his 2.2 and 2.3 use most of what is in my proof-sketches (formal groups, quasi-finite flat groups, Raynaud's theorem, etc.). | |
| Nov 5, 2012 at 17:20 | comment | added | Tim Dokchitser | Perhaps Vatsal is simply referring to a theorem of Serre that if $\phi$ is an $l$-isogeny between semistable elliptic curves over $\mathbb Q$ (all assumptions necessary), then there are only two possibilities: either $\ker\phi\cong{\mathbb Z}/l{\mathbb Z}$ or $\ker\phi\cong\mu_l$. [Proprietes galoisiennes..., Invent. Math. 15 (1972), Section 5.4, Lemme 6 and below]. Taking the dual isogeny swaps the two; $l=2$ is allowed. | |
| Nov 4, 2012 at 21:59 | comment | added | user27056 | @Will: In view of the OP's apparent background, a genuine answer would have to delve into rather more precise detail (such as the link between Tate uniformization and Neron models, between formal groups and identity components of $p$-divisible groups, the statements of the theorems of Grothendieck and Raynaud being invoked, interaction of Neron model with various kinds of base change). I don't feel like getting into all of that here or chasing down literature references, and the "comment" section is more appropriate for sketches. Anyone who wishes is welcome to expand it into a proper answer. | |
| Nov 4, 2012 at 19:14 | comment | added | Will Sawin | I think you should post all that as an answer so Taekyung Kim can accept it. | |
| Nov 4, 2012 at 18:00 | comment | added | user27056 |
@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).
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| Nov 4, 2012 at 17:53 | comment | added | Keenan Kidwell | I should have said finite flat $\mathbf{Z}_\ell$-group schemes of $\ell$-power order. | |
| Nov 4, 2012 at 17:50 | comment | added | Keenan Kidwell | For example, because their special fibers are not isomorphic. | |
| Nov 4, 2012 at 17:47 | comment | added | Keenan Kidwell | Dear @Will, I think the problem with 2 is that the following theorem of Raynaud fails: the functor $H\rightsquigarrow H(\overline{\mathbf{Q}}_\ell)$ from finite flat $\mathbf{Z}_\ell$-group schemes to "finite flat" $G_{\mathbf{Q}_\ell}$-modules is fully faithful. In general this applies to extensions of $\mathbf{Q}_\ell$ with ramification index less than $\ell-1$, so $2$ is ruled out. I think an example where it fails for $\ell=2$ comes from $\mu_2$ and $(\mathbf{Z}/2\mathbf{Z})_{\mathbf{Z}_\ell}$, which have the same generic fiber but are not isomorphic. | |
| Nov 4, 2012 at 17:40 | comment | added | Will Sawin | What's the counterexample for $l=2$? | |
| Nov 4, 2012 at 17:25 | comment | added | user27056 | Just as an aside, your $N(\phi)$ is almost never finite or surjective, so you should not call it an "isogeny". (In the relative setting, you should think carefully about what deserves to be called an "isogeny" and why; i.e., what properties this should satisfy so that it exhibits behavior as pleasant and useful as in the theory over a field with smooth groups of finite type that may not be connected.) | |
| Nov 4, 2012 at 17:09 | comment | added | user27056 | 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 | |
| Nov 4, 2012 at 16:42 | comment | added | user27056 | 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. | |
| Nov 4, 2012 at 13:55 | comment | added | R.P. | "We say that φ is étale if the extension [...] to Néron models is étale." | |
| Nov 4, 2012 at 13:47 | comment | added | Felipe Voloch | He must be using some non-standard definition of etale. | |
| Nov 4, 2012 at 13:07 | history | asked | Taekyung Kim | CC BY-SA 3.0 |