Timeline for Ramification behavior of field given by adjoining $p$-torsion point on formal group of abelian variety
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13 events
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| Apr 8, 2021 at 2:56 | comment | added | naf | It seems likely to me that any lift (over the Witt vectors) of a (principally polarised) abelian suface $A$ over $\overline{\mathbb{F}}_p$ such that $A[p]$ is connected but not isomorphic to $E[p]^2$, where $E$ is a supersingular elliptic curve, will have $p$-torsion which is not tamely ramified. This should be provable using Raynaud's results mentioned in @WillSawin's comment, but I have not checked carefully. | |
| Apr 8, 2021 at 0:40 | comment | added | naf | A reference for the embedding theorem of Raynaud is Theorem 3.1.1 of "Theorie de Dieudonne cristalline II" by Berthelot, Breen and Messing. Also, it should probably be easy to construct non-semisimple group schemes using Breuil modules. | |
| Apr 7, 2021 at 21:49 | comment | added | Will Sawin | For Raynaud's classification, I think "Finite Flat Group Schemes" by Tate in the book Modular Forms and Fermat's Last Theorem link.springer.com/chapter/10.1007/978-1-4612-1974-3_5 is a good introduction. For semisimplicity, it's the classical fact that representations of a group of order prime to p are semisimple in characteristic p. | |
| Apr 7, 2021 at 21:10 | comment | added | Jackson Morrow | @WillSawin Thank you for the comment! This looks like a great avenue to explore to get a counterexample. Could you provide some references to the statements about tameness implying semi-simplicity and Raynaud's classification of simple finite flat group schemes? | |
| Apr 7, 2021 at 20:59 | comment | added | Jackson Morrow | @naf Thank you for the comment! Could you provide me with a reference to this theorem of Raynaud? It looks like a very relevant resource. | |
| Apr 7, 2021 at 16:10 | comment | added | Will Sawin | Finite flat group schemes over such a ring are determined by their Galois representation. Tameness implies that the Galois representation is semisimple. Since subrepresentations of the Galois representation of a finite flat group scheme come finite flat subgroup schemes, tameness implies that the finite flat group scheme is a product of simple finite flat group schemes. These simple finite flat group schemes were classified (also by Raynaud). So any finite flat group scheme that is not a sum of these would be an example. | |
| Apr 7, 2021 at 14:28 | comment | added | naf | By a theorem of Raynaud, any finite flat group scheme, say over a dvr, can be embedded in an abelian scheme. So your question amounts to asking whether points of order $p$ of a finite flat group sheme with connected special fibre generate a tamely ramified extension. This seems unlikely to me. | |
| Apr 7, 2021 at 12:30 | comment | added | Will Sawin | Sorry, my mistake. | |
| Apr 7, 2021 at 10:25 | comment | added | Jackson Morrow | @WillSawin Thanks for the comment! Yes, I did mean to assume that the $p$-torsion lies in the formal group, which is why I wrote $P$ is in $\widehat{\mathcal{A}}(\mathcal{O}_{\overline{K}})$. Also, I want to assume the abelian variety has good reduction, so not a Tate curve. Your example is nice and as you said will give wild ramification but it's not the situation I am interested in. | |
| Apr 7, 2021 at 0:36 | history | edited | Jackson Morrow |
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| Apr 5, 2021 at 21:44 | comment | added | Jackson Morrow | @Lubin Thank you for the comment! Of course, you are totally correct that there isn't anything fancy going on in the supersingular curve case over an unramified base other than the theory of Newton polygons. Moreover, there really wasn't any need to add the citation to J.-P. Serre's work. I just learned of the result from that work so I thought I would include it. | |
| Apr 5, 2021 at 21:11 | comment | added | Lubin | In the case of supersingular curves over an unramified base, there’s nothing fancy going on, to see that the ramification is tame. You see that the Newton Polygon of $[p]$ has vertices at $(1,1)$ and $(p^2,0)$, so all $p$-torsion points have valuation $1/(p^2-1)$. Naturally, all bets are off in cases where there’s ramification in the base. | |
| Apr 5, 2021 at 19:49 | history | asked | Jackson Morrow | CC BY-SA 4.0 |