Timeline for components of E[p], E universal in char p.
Current License: CC BY-SA 2.5
16 events
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| Mar 11, 2010 at 21:12 | comment | added | BCnrd | @Tyler: I agree it's the same as a "complete set of roots" (defined in one of several equivalent ways). What I didn't recall offhand was if this is captured just by dividing the equations or needs more conditions. For $\mu_N$ stuff it is just dividing, but for more complicated group schemes I just didn't remember. | |
| Mar 11, 2010 at 19:13 | comment | added | Tyler Lawson | @Brian: I don't have Katz-Mazur handy but looking at Cor. II.2.3 in Harris-Taylor seems to be essentially establishing that a full Drinfeld level m structure (in this context) is basically the same as a complete set of roots of the equation defining p^m-torsion, at least in the one-dimensional case. They claim that this was Drinfeld's original approach and KM use a slight variation. My apologies, I'm not a specialist in this area but I am not sure what is different about the supersingular case - I'd like to know what your source of skepticism is. | |
| Mar 11, 2010 at 18:13 | comment | added | BCnrd | @Tyler: Oh, so "subtraction" in the sense of Cartier divisors? It is tantamount to saying that a "point of exact order $p^n$" in the sense of KM is the same as "a $p^n$-torsion point that is not $p^{n-1}$-torsion". That works along the ordinary part, and gives exactly what Katz-Mazur do (with a small argument). I am doubtful it gives the right notion at the ss points; that is where all of the real content lies (since torsion in the ordinary case is sort of "easy" to understand). | |
| Mar 11, 2010 at 17:24 | comment | added | Tyler Lawson | @Brian: I think the point is that he's not actually localizing at the identity section, but instead dividing the equation defining the p-torsion by the equation defining the identity section - is this roughly correct? | |
| Mar 11, 2010 at 17:22 | comment | added | BCnrd | Kevin, I don't understand your proposed construction. By removing the underlying space of the identity section from E[p] you're removing the whole space at the ss points. So why should it not ruin everything there? The construction in KM is sort of"explicit" about how the coordinate rings are made (e.g., they're not appealing to abstract theorems of Artin, or Hilbert schemes, etc.), and I don't see how totally explicit equations help to do anything. | |
| Mar 11, 2010 at 16:54 | comment | added | Kevin Buzzard | @Tyler: I had always assumed Drinfeld's contribution was to construct the scheme. Now I'm wondering whether there is a natural construction of the scheme, and Drinfeld's contribution was to note that it represented a "points of order p^n" functor. Maybe E[p^n] - E[p^{n-1}] works to give Drinfeld X_1(p^n) too. | |
| Mar 11, 2010 at 16:53 | comment | added | Kevin Buzzard | Brian: I think the "E[p] minus origin" might even work at the ss points. Here's how I want to construct Drinfeld Y_1(p) (following Rezk and omitting an auxiliary tame level structure if you want to stay with schemes): E[p] is an effective Cartier divisor in E/Y_0(1), and it contains the zero section, so E[p] - (zero-section) is an effective Cartier divisor. The associated scheme seems to me to be Y_1(p) (over Z). Do you buy this? | |
| Mar 11, 2010 at 14:48 | comment | added | BCnrd | @Tyler, @Kevin: I think the real magic is that the Drinfeld structure stuff works at the supersingular points too, for which there's no analogue of the question as asked. Kevin, for answers to the two puzzles see Example A.8.3 in "Pseudo-reductive groups". (Now I see another little non-math typo there to be fixed...) Maybe you'll also like the first few paragraphs of section A.6 there. Just reconfirms how disorienting imperfect fields case be. :) | |
| Mar 11, 2010 at 13:22 | comment | added | Tyler Lawson | @Kevin: You suggest that perhaps this doesn't work for level p^n-structures - does $E[p^n] \setminus E[p^{n-1}]$ have some properties different from $E[p] \setminus E$? | |
| Mar 11, 2010 at 11:56 | vote | accept | Kevin Buzzard | ||
| Mar 11, 2010 at 11:56 | comment | added | Kevin Buzzard | ...whereas maybe this trick doesn't? Or perhaps it's not so great because you don't get a moduli-theoretic interpretation of the answer (i.e. perhaps Drinfeld's insight was to interpret this space as a moduli problem, rather than constructing the space), but I think that it's a cheap and dirty way of constructing the representing object that works over Z. | |
| Mar 11, 2010 at 11:54 | comment | added | Kevin Buzzard | OK so here's the reason I was interested (I could have put this in the question but it was already long enough). Rezk wanted to construct explicit equations for Drinfeld Gamma_1(p) structures, as you know. He wanted to do this by writing down the equation for E[p] and dividing by the equation for the identity section. I was very skeptical! If it were as simple as this, why make so much fuss about Drinfeld level structures! Deligne-Rapoport could have done this in the 70s. But I think it works! Perhaps the key fact about Drinfeld level structures is that they work for level p^n whereas... | |
| Mar 11, 2010 at 11:52 | comment | added | Kevin Buzzard | Brian this is very nice---thanks. In fact your answer does everything: (a) it illuminates why my "gut instinct" breaks down (non-reducedness isn't local for the fppf topology) and (b) it saves me from having to do the deformation theory calculation. You don't need to introduce y by the way; you can just change T to S=T-1 and it's already Eisenstein over k[[x]] for i!=0 (the case i=0 of course corresponds to the identity component so this argument makes it manifestly clear that the identity component is behaving differently). I can't do either of your puzzles offhand :-/ | |
| Mar 11, 2010 at 2:57 | history | edited | BCnrd | CC BY-SA 2.5 |
added 12 characters in body
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| Mar 11, 2010 at 2:37 | history | edited | BCnrd | CC BY-SA 2.5 |
I made the Eisenstein argument a little bit clearer
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| Mar 11, 2010 at 0:27 | history | answered | BCnrd | CC BY-SA 2.5 |