# The Shafarevich Conjecture and motivic Langlands stacks.

Hi, I recently learned about an amazing conjecture of Shafarevich (proved by Faltings) about the finiteness of the number of curves of a fixed genus with good reduction outside a finite number of primes. Moreover, there are no curves of genus $> 0$ at all with good reduction everywhere (Abrashkin/Fontaine).

Is there an analog of this theorem for stacks (of the type described below), and how does this connect to the Langlands program?

From what I understand, the moduli space $\mathcal{M}_{1,1}$ of elliptic curves can be thought of a stack with good reduction everywhere over $\mathbb{Z}$. (Mumford computed that the Picard group of this stack is $\mathbf{Z}/12\mathbf{Z}$.) Moreover, the generic fiber of this stack of the form $[X/G]$, where $X/\mathbf{Q}$ is a smooth proper curve and $G$ is a finite group. I want to restrict attention to exactly this special class of stacks (do they have a name?).

First question: can one classify smooth proper stacks $\mathcal{X}$ over $\mathbf{Z}$ with generic fiber $[X/G]$ for some smooth proper curve $X$ over $\mathbf{Q}$ and finite group $G$? Are there finitely many such stacks? Is $\mathcal{M}_{1,1}$ the only one with negative euler characteristic?

Second question: are there finitely many smooth stacks $\mathcal{X}$ over $\mathbf{Z}[1/N]$ where $N$ and $\chi(\mathcal{X})$ are fixed?

Finally, is there any Tannakian/Langlands/Motivic formulism that attaches some motivic type object to $\mathcal{M}_{1,1}$ that isn't just the "trivial" motive attached to $\mathbf{P}^1$?

Apologies for any vagueness in this question, hopefully a more seasoned MO Langlands pro like David Hansen or James Taylor can help me out.

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Dear Professor Dalawat - thanks for the links! This is addressing the last question I presume? If I understand you, you are saying that the "1-motive" should be an abelian variety over $\mathbf{Z}$ which can't exist. But I was thinking along the following lines: is there some integral "Motivic like" object $[\mathcal{M}]$ for which "$H^1$" returned $\mathbf{Z}/\mathbf{Z}12$. I don't even know if Motives are supposed to form a category with "integral" properties... and maybe this is related to "torsion" automorphic forms... perhaps your remarks merely expose my ignorance, but I like to dream! –  Spaghetti Inks Nov 29 '11 at 4:10
I provided the links merely because they discuss the relationship of the Langlands' Programme with Fontaine-Abrashkin. –  Chandan Singh Dalawat Nov 29 '11 at 6:00
(If the "Worst notation ever" question was still open, I would add the notation " $X/k$ " often used by algebraic geometers to mean "the scheme $X$ is defined over $k$"!) –  Qfwfq Nov 29 '11 at 10:07