Timeline for Models of the modular curve $Y_1(N)$
Current License: CC BY-SA 3.0
7 events
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| Nov 20, 2012 at 17:34 | comment | added | François Brunault | @crocodile : I'm not familiar with the language of "stacks" used by Deligne-Rapoport but I think you're right. The isomorphism $w : \mathcal{A}_N \to \mathcal{B}_N$ in Deligne-Rapoport is none other than the Atkin-Lehner involution on $X_0(N)$ (or rather its model over $\mathbf{Z}[1/N]$). | |
| Nov 20, 2012 at 11:45 | comment | added | crocodile | If I've understood Deligne--Rapoport correctly, if $N$ is invertible on the base the moduli problems $\mathcal{A}_N$ and $\mathcal{B}_N$ are identical, aren't they? When $N$ is invertible, $\mu_N$ is etale-locally isomorphic to $\mathbb{Z} / N$ (they become isomorphic after base-change to $\mathbb{Z}[1/N, \zeta_N]$, which is \'etale over $\mathbb{Z}[1/N]$). At least, that would explain why Deligne and Rapoport only mention them in the chapter "Reduction modulo p". | |
| Nov 19, 2012 at 8:59 | vote | accept | crocodile | ||
| Nov 17, 2012 at 9:10 | comment | added | François Brunault | @stankewicz : I don't think the $Y_\mu(N)$ described here is a model for $Y_0(N)$. I really consider the closed immersion as part of the data, not only its image in $E$, like the $\mathcal{A}_N$ and $\mathcal{B}_N$ moduli problems in Deligne-Rapoport. Two closed immersions $i_1 : \mu_N \to E_1$ and $i_2 : \mu_N \to E_2$ are equivalent when there is an isomorphism $psi : E_1 \xrightarrow{\cong} E_2$ such that $i_2 = \psi \circ i_1$. Any $\alpha \in ({\bf Z}/N{\bf Z})^\times$ defines an isomorphism $[\alpha] : \mu_N \to \mu_N$, but in general $(E,i) \not\cong (E,i \circ [\alpha]$. | |
| Nov 16, 2012 at 21:45 | comment | added | stankewicz | I think your $Y_\mu(N)$ is a model for $Y_0(N)$. For one thing, $Y_1(N)$ parametrizes elliptic curves with a point of exact order N, not with a cyclic subgroup of order $N$ as $\mu_N$ is. The two moduli problems for $Z/N$ embeddings and $\mu_N$ embeddings are the $\mathcal{A}_N$ and $\mathcal{B}_N$ moduli problems in Deligne-Rapoport's section V.1. | |
| Nov 16, 2012 at 21:17 | history | edited | François Brunault | CC BY-SA 3.0 |
added 249 characters in body
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| Nov 16, 2012 at 21:07 | history | answered | François Brunault | CC BY-SA 3.0 |