Timeline for How do you explicitly compute the p-torsion points on a general elliptic curve in Weierstrass form?
Current License: CC BY-SA 2.5
13 events
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| Mar 11, 2010 at 0:05 | comment | added | Kevin Buzzard | @Charles: I posted some of my thoughts at mathoverflow.net/questions/17782/… | |
| Mar 10, 2010 at 23:20 | comment | added | Kevin Buzzard | The upshot of our email conversation was that my gut instinct above is wrong. I claim above that above the ordinary locus of the mod p modular stack, E[p] (finite flat over the stack and hence an abstract algebraic curve) has no reduced components. The reason I thought this was because its identity component is non-reduced, and "hence all the other components must be too". I can't prove this, and I am convinced by Charles' arguments, so my slip must be here. It seems to me that one can form Drinfeld Gamma_1(p) by throwing out the identity section of E[p]. My apologies to Charles. | |
| Mar 9, 2010 at 15:03 | comment | added | Charles Rezk | I think I'm being dense about what I mean about "reduced". Let's continue this in email. | |
| Mar 9, 2010 at 12:45 | comment | added | Kevin Buzzard | @Charles: actually I am too hasty suggesting you have $p$ components because over a big base they might link up. I should localise and complete at a point on the base first. | |
| Mar 9, 2010 at 11:49 | comment | added | Kevin Buzzard | Another remark: this question has drifted down the list of questions and hence I am unlikely to notice if you respond to my comments above---the question only gets bumped to the top if someone posts a new answer, not a new comment. Hence if you want to alert me to any comments you have, we could continue the discussion by email. | |
| Mar 9, 2010 at 11:46 | comment | added | Kevin Buzzard | As for your argument in the comments above, I don't know what you mean by "$\Gamma_1(m)$ is a closed subscheme of...". Are you confusing a $\Gamma_1(m)$ structure with the moduli space representing $\Gamma_1(m)$ structures? | |
| Mar 9, 2010 at 11:44 | comment | added | Kevin Buzzard | @Charles: I really think this is no good. You seem to be suggesting that $E[p]$ minus the zero section is representing Drinfeld $\Gamma_1(p)$ structures. But here's a proof that this can't be true: consider a "universal" connected family of ordinary elliptic curves in characteristic $p$ (e.g. the ord locus of $X_1(N)$ mod p for some auxiliary $N>4$ prime to $p$). The object you suggest is representing the functor has $p$ components and none of them are reduced. The object that does represent the functor, according to the table on p417 of Katz-Mazur, has two components, one of which is reduced. | |
| Mar 7, 2010 at 19:25 | comment | added | Charles Rezk | Well, K-M define a $\Gamma_1(m)$ structure as a group homomorphism $f:Z/m\to E$ (equivalently, a choice of $f(1)\in E[m]$) which satisfies a certain condition (an inequality $\sum [f(i)]\leq E[m]$ of effective divisors). This is a closed condition, so $\Gamma_1(m)$ is a closed subscheme of $E[m]/S$; I believe they show it is flat and finite over the base $S$ (KM 5.1.1), of rank $\#\text{injections}(Z/m,Z/m^2)$. For $m=p$, this should do it, no? | |
| Mar 7, 2010 at 15:42 | comment | added | Kevin Buzzard | Well perhaps I am confused as well; I'll try and remember to look at my copy of K-M on Monday (you seem to indicate that I've forgotten what the special fibre of X_1(p) looks like, which might be true; I had thought it had two components, one a (reduced) Igusa curve and one non-reduced). But my point is this. Why should E[p] minus the origin represent points of order p in the sense of Drinfeld? If that's true (which it might be) then it's news to me. I'll try to remember to get back to you. If this trick works then you're in business! | |
| Mar 6, 2010 at 22:22 | comment | added | Charles Rezk | Perhaps I am confused, or am using incorrect terminology. The gadget I want should not have reduced components in char p; I want a Gamma_1(p) structure in the sense of Katz-Mazur, 3.2. | |
| Mar 6, 2010 at 19:35 | comment | added | Kevin Buzzard | If you're happy with $E[p]$ minus the identity then great. That sort of notion was understood in the 1960s. Drinfeld level structures are more subtle and weren't around until the 1980s. | |
| Mar 6, 2010 at 19:34 | comment | added | Kevin Buzzard | What you seem to be doing is writing down the divisor $E[p]$ minus the identity section. I thought you wanted the scheme parametrising points of order $p$ in the sense of Drinfeld. The problem is that in characteristic $p$ the relationship between these two things is delicate. For example $E[p]$ minus the identity, over an ordinary elliptic curve over an alg closed field of char $p$, will typically have all its components non-reduced. But the moduli space for the Drinfeld level structure will have reduced components IIRC. I am unclear about what you want though. | |
| Mar 6, 2010 at 18:46 | history | answered | Charles Rezk | CC BY-SA 2.5 |