Stacks in modern number theory/arithmetic geometry 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.
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). 
I asked this question at M.SE (here : https://math.stackexchange.com/questions/143746/stacks-in-arithmetic-geometry 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.
EDIT: I have belatedly made this question CW, as I've realised, too late, that there is clearly not one correct answer.
 A: One big recent example would be Lafforgue's proof of the Langlands correspondence for $GL_n$ of function fields (http://arxiv.org/abs/math.NT/0212399), 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.  
More readable versions, with additional context, are given by Lafforgue's advisor Gerard Laumon (http://arxiv.org/abs/math.AG/0003131 if you can read French) and by his student Ngo Dac Tuan (MR2402699 on MathSciNet, or http://www.impan.pl/~pragacz/download/Ngo.pdf)
A: Here are two applications of stacks to number theory.
1) Section 3 of this paper, which solves the diophantine equation $x^2 + y^3 = z^7$, explains the connection between stacks and generalized Fermat equations.
2) This post explains how stacks fit into the proof of Deuring's formula for the number of supersingular elliptic curves over a finite field.
A: 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.
