User spaghetti inks - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T09:40:31Z http://mathoverflow.net/feeds/user/19595 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/82131/the-shafarevich-conjecture-and-motivic-langlands-stacks The Shafarevich Conjecture and motivic Langlands stacks. Spaghetti Inks 2011-11-29T01:40:07Z 2011-11-29T01:40:07Z <p>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).</p> <p>Is there an analog of this theorem for stacks (of the type described below), and how does this connect to the Langlands program?</p> <p>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?).</p> <p>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?</p> <p>Second question: are there finitely many smooth stacks $\mathcal{X}$ over $\mathbf{Z}[1/N]$ where $N$ and $\chi(\mathcal{X})$ are fixed?</p> <p>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 $\mathbf{P}^1$?</p> <p>Apologies for any vagueness in this question, hopefully a more seasoned MO Langlands pro like David Hansen or James Taylor can help me out.</p> http://mathoverflow.net/questions/82131/the-shafarevich-conjecture-and-motivic-langlands-stacks Comment by Spaghetti Inks Spaghetti Inks 2011-11-30T04:49:38Z 2011-11-30T04:49:38Z Naturally $X/k$ is the notation, I say to old Qfwfq, what else could it have been? http://mathoverflow.net/questions/82131/the-shafarevich-conjecture-and-motivic-langlands-stacks Comment by Spaghetti Inks Spaghetti Inks 2011-11-30T04:48:03Z 2011-11-30T04:48:03Z Dear Professor Carnahan, I would love it if you could explain the construction (&quot;it is not too difficult to make genus zero curves with .. good reudction everywhere&quot;). http://mathoverflow.net/questions/82131/the-shafarevich-conjecture-and-motivic-langlands-stacks Comment by Spaghetti Inks Spaghetti Inks 2011-11-29T04:10:17Z 2011-11-29T04:10:17Z Dear Professor Dalawat - thanks for the links! This is addressing the last question I presume? If I understand you, you are saying that the &quot;1-motive&quot; should be an abelian variety over $\mathbf{Z}$ which can't exist. But I was thinking along the following lines: is there some integral &quot;Motivic like&quot; object $[\mathcal{M}]$ for which &quot;$H^1$&quot; returned $\mathbf{Z}/\mathbf{Z}12$. I don't even know if Motives are supposed to form a category with &quot;integral&quot; properties... and maybe this is related to &quot;torsion&quot; automorphic forms... perhaps your remarks merely expose my ignorance, but I like to dream!