most recent 30 from http://mathoverflow.net 2013-06-19T11:55:01Z http://mathoverflow.net/feeds/question/96957 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/96957/stacks-in-modern-number-theory-arithmetic-geometry David Roberts 2012-05-15T00:45:13Z 2012-05-15T05:50:12Z

<p>Stacks, of varying kinds, appear in algebraic geometry whenever we have moduli problems, most famously the stacks of (marked) curves. But these seem to be to be very geometric in motivation, so I was wondering if there are natural examples of stacks that arise in arithmetic geometry or number theory.</p> <p>To me, as a non-expert, it seems like it's all Galois representations and estimates on various numeric measures (counting points or dimensions of things) based on other numeric things (conductors, heights, etc). </p> <p>I asked this question at M.SE (here : <a href="http://math.stackexchange.com/questions/143746/stacks-in-arithmetic-geometry" rel="nofollow">http://math.stackexchange.com/questions/143746/stacks-in-arithmetic-geometry</a> please vote to close if you can) because I thought it a bit too 'recreational', but with no success. What I am after is not just stacks which can be seen as arithmetic using number fields or rings of integers, but which are actually used in number-theoretic problems, or have a number-theoretic origin. Maybe there aren't any, but it doesn't hurt to ask.</p> <p>EDIT: I have belatedly made this question CW, as I've realised, too late, that there is clearly not one correct answer.</p>

http://mathoverflow.net/questions/96957/stacks-in-modern-number-theory-arithmetic-geometry/96961#96961 Zack Wolske 2012-05-15T03:13:40Z 2012-05-15T04:30:51Z

<p>One big recent example would be Lafforgue's proof of the Langlands correspondence for $GL_n$ of function fields (<a href="http://arxiv.org/abs/math.NT/0212399" rel="nofollow">http://arxiv.org/abs/math.NT/0212399</a>), which uses stacks of schtukas. It is similar to Drinfel'd's proof for $GL_2$, but with the moduli space being an essential component. </p> <p>More readable versions, with additional context, are given by Lafforgue's advisor Gerard Laumon (<a href="http://arxiv.org/abs/math.AG/0003131" rel="nofollow">http://arxiv.org/abs/math.AG/0003131</a> if you can read French) and by his student Ngo Dac Tuan (MR2402699 on MathSciNet, or <a href="http://www.impan.pl/~pragacz/download/Ngo.pdf" rel="nofollow">http://www.impan.pl/~pragacz/download/Ngo.pdf</a>)</p>

http://mathoverflow.net/questions/96957/stacks-in-modern-number-theory-arithmetic-geometry/96962#96962 stankewicz 2012-05-15T03:51:16Z 2012-05-15T03:51:16Z

<p>Perhaps this doesn't count as "modern" but stacks are ubiquitous in the 1972 Antwerp paper of Deligne and Rapoport. Recall that the $\Gamma_0(N)$ moduli problem is not representable, and so they must frequently work directly with stacks before moving to the coarse moduli scheme we all know and love.</p>

http://mathoverflow.net/questions/96957/stacks-in-modern-number-theory-arithmetic-geometry/96968#96968 David Zureick-Brown 2012-05-15T05:45:04Z 2012-05-15T05:45:04Z

<p>Here are two applications of stacks to number theory.</p> <p>1) Section 3 of <a href="http://math.mit.edu/~poonen/papers/pss.pdf" rel="nofollow">this</a> paper, which solves the diophantine equation $x^2 + y^3 = z^7$, explains the connection between stacks and generalized Fermat equations.</p> <p>2) <a href="http://mathoverflow.net/questions/24573/" rel="nofollow">This</a> post explains how stacks fit into the proof of Deuring's formula for the number of supersingular elliptic curves over a finite field.</p>