Timeline for A reference for $\mathbb{A}^1_R$ being a coarse moduli space of the stack of elliptic curves
Current License: CC BY-SA 3.0
13 events
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| Aug 31, 2017 at 20:51 | answer | added | Minseon Shin | timeline score: 3 | |
| Jul 3, 2016 at 18:12 | comment | added | Lennart Meier | Section 3.3. of arxiv.org/pdf/1511.07475.pdf is a reference in the compactified case. | |
| Mar 9, 2015 at 7:00 | comment | added | user74230 | @QuestionMark: The only method which works is to use Raynaud's construction via formal schemes as developed in Deligne-Rapoport (and so it goes through generalized elliptic curves!). Of course, one might want to then prove that Raynaud's construction actually coincides with the usual Tate curve, another fact you'll never find proved in the literature (but is a nice exercise with formal GAGA and inspection of Tate's classical computations), but actually this agreement is not so important since the explicit equation is basically useless (though has psychological value). | |
| Mar 9, 2015 at 5:18 | comment | added | Question Mark | @user74230: Regarding the reference to Roquette's book that you've alluded to, what would be your suggestion for handling properties of the Tate curve needed in 8.8 of K-M? As far as I can tell none of the references indicated in K-M fully justify the claims made in 8.8. These claims seem believable, but is there a reference which could be used to verify them? | |
| Mar 6, 2015 at 17:42 | comment | added | Question Mark | @user74230: I don't think that more detailed explanations are needed, since your comments fully answer the question, but it's up to you, of course, whether to upgrade them to an answer. Thanks for your help. | |
| Mar 6, 2015 at 2:34 | comment | added | user74230 | @QuestionMark: I don't feel like writing detailed explanations to make a proper "answer"; if someone else cares then they're welcome to do that (but then should remove appeal to regularity, using that $A/A^G$ is torsion-free; it is the smoothness of $A^G$ in this case that is the more important aspect). The categorical property of Spec($A^G$) (say for $A$ of finite type over a noetherian ring $R$ and $G$ acting over $R$) relative to algebraic spaces is an exercise with the etale topology and monicity of diagonal maps. | |
| Mar 5, 2015 at 22:38 | comment | added | Question Mark | @user74230: Thanks! You should post this as an answer. Katz and Mazur don't mention this, but the coarse moduli spaces they define in (8.1.1) are actually coarse moduli spaces in the Deligne-Rapoport sense due to (8.1.3), at least if one restricts to morphisms towards schemes, right? Is the latter scheme restriction necessary? Is $\mathrm{Spec}(A^G)$ still the categorical quotient in the category of algebraic spaces? | |
| Mar 5, 2015 at 4:08 | comment | added | user74230 | The reference in K-M to Igusa's paper is as mystifying as their reference to Roquette's book on p-adic analysis (or anything other than D-R) for proving anything serious about the structure of the Tate curve over $\mathbf{Z}$ (such as structure of its $n$-torsion group schemes). I did not find such references to help in logical understanding of anything (but of historical interest, to see how people had to struggle to express deep ideas in earlier times). | |
| Mar 5, 2015 at 4:05 | comment | added | user74230 | For the coordinate ring $A$ of $Y(p)$ over $\mathbf{Z}[1/p]$ and prime $p\ge 3$ and $G={\rm{GL}}_2(\mathbf{F}_p)$, we want $R\otimes A^G \rightarrow (R\otimes A)^G$ is an equality. We know $A^G=\mathbf{Z}[j][1/p]$ by D-R, so $A^G\rightarrow A$ is finite flat by regularity stuff, so injectivity holds for any $R$. But $R$ matters through its underlying abelian group, so (via direct limits) for surjectivity we reduce to $R=\mathbf{Z}/n\mathbf{Z}$ and then $\mathbf{F}_{\ell}$ for prime $\ell\ne p$. But the finite flat $j$-map on $Y(p)_{\mathbf{F}_{\ell}}$ has degree $\#G$, so we are done. QED | |
| Mar 4, 2015 at 20:20 | answer | added | Lennart Meier | timeline score: 4 | |
| Mar 4, 2015 at 16:22 | comment | added | Question Mark | Yes, I have looked into that. Unfortunately, they don't prove it there. They say in (8.2.1) that this is "well-known" and they give a reference to some paper of Igusa's (unfortunately, I may need a time machine to properly understand the latter, but I guess that the property justified there is bijectivity on $\overline{k}$-points). Katz and Mazur do, however, base their constructions on the $j$-line, but perhaps the case $R = \mathbb{Z}$ is the most important one (where Deligne-Rapoport applies), or perhaps they get around it with their ad hoc definition of a coarse moduli space in 8.1.1. | |
| Mar 4, 2015 at 9:07 | comment | added | Daniel Loughran | I don't have it with me, so I'm not sure if it's in there, but have you tried looking in Katz, Mazur - Arithmetic Moduli of Elliptic Curves ? | |
| Mar 4, 2015 at 3:10 | history | asked | Question Mark | CC BY-SA 3.0 |