Will Chen
  • Member for 10 years, 8 months
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  • New York, NY
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For your first question, your use of $SL(2,\mathbb{Z})(\widehat{\mathbb{Z}})$ is confusing, since in the rest of your post you seem to use $SL(2,\mathbb{Z})$ as a group, not a group scheme. If you really meant $SL(2,\widehat{\mathbb{Z}})$, then the relation between that and $\widehat{SL(2,\mathbb{Z}))$ is that the latter surjects onto the former (by the universal property of profinite completions), the kernel being free profinite of countable rank.

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@DamianRössler I don't think prestable curves are projective (or quasi-projective). If they were, then you'd have a relatively ample sheaf, so Zariski locally on the base, $C$ would satisfy (AF) by EGA II Corollaire 4.5.4, which is enough for the quotient to be a scheme.

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I had long wondered this myself! Note that if true, it would realize the Suzuki group of order 29120 as a Galois group over $\mathbb{Q}$ (in the sense of the inverse Galois problem). This would probably also work for a number of other groups. See for example $\S4.3$ in arxiv.org/pdf/1510.05687.pdf

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$\Gamma(2)$ is actually not free. For example it contains $-I$ (this is a common mistake). However there is a subgroup of $\Gamma(2)$ which is free, which is sometimes called the Sanov subgroup.

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@SamanthaSmith Stacks are fibered categories. Principal bundles over schemes are schemes. Principal bundles over stacks are stacks. Yes principal bundles can also be given on an atlas, but that data also defines an honest stack over $M_{G,C}$, which you might call the "total space" of the principal bundle.

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Let $P_{univ}$ denote this universal bundle. I believe the correct definition of $P_{univ}$ is this: It is the fibered category whose objects over a scheme $S$ are pairs $(P,s)$ where $P$ is a principal $G$-bundle over $C\times S$, and $s$ is a section of $P$, and morphisms are morphisms of bundles preserving the section. The forgetful map to $M_{G,C}$ is visibly representable, and one checks that the pullback of $P_{univ}$ via any map $f : S\rightarrow M_{G,C}$ is precisely the bundle corresponding to $f$. This is probably what Sorger means.

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SGA 1

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@user237522 I don't understand your question

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