Timeline for What can we do with a coarse moduli space that we can't do with a DM moduli stack?
Current License: CC BY-SA 2.5
6 events
| when toggle format | what | by | license | comment | |
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| Apr 1, 2010 at 17:36 | vote | accept | Kevin H. Lin | ||
| Mar 13, 2010 at 17:57 | comment | added | Kevin Buzzard | Oh my goodness yes of course the special fibre of the Neron model of the Jacobian is controlled by the reduction of the curve and of course it sees all of those e's doesn't it. So in fact this makes James Borger's comment even more bewildering: "you wouldn't think it comes up too often" but in fact those e's explain all sorts of things, like "dihedral" (non-Eisenstein) errors in the theory of mod p modular forms and so on (e.g. the level 13 counterexample to Serre's conjecture if you demand that the char of the char 0 form is the Teichmuller lift of the mod p form). | |
| Mar 13, 2010 at 17:09 | comment | added | BCnrd | @Kevin: I think it would be awkward at best, and probably impossible to pull off. For starters, he makes essential use the Neron model of the Jacobian, which is influenced by the structure of the minimal regular resolution, whereas the moduli stack is regular (and it seems unlikely that a non-scheme stack would give rise to a "Jacobian" with useful properties like for the coarse scheme). Also, since $X_0(p)$ is "nowhere" a scheme (even over $\mathbf{Q}$) and there's no link to rational points on $Y_1(N,p)$ with $N \ge 4$, extra level structure doesn't seem to help. | |
| Mar 13, 2010 at 16:46 | comment | added | Kevin Buzzard | @Brian: Mazur's paper is a great thing to bring up. Stacky Y_0(p) isn't a scheme, but a lot of time is spent in that paper analysing scheme Y_0(p) (and of course the local equations at the ss points are different: XY=p vs XY=p^e). So in some sense the stack and the scheme really are quite different objects (the scheme isn't always even regular!). I wonder what happens if you "try to prove Mazur's theorem using only stacks", i.e. you're not "allowed to use" the scheme Y_0(p) with local equation XY=p^e at the ss points, but you can have the stack and also the scheme Y_1(N;p) for N>=4 etc. | |
| Mar 13, 2010 at 16:38 | history | edited | BCnrd | CC BY-SA 2.5 |
added 539 characters in body
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| Mar 13, 2010 at 16:11 | history | answered | BCnrd | CC BY-SA 2.5 |