# 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 : http://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.

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I'm a little confused: Are you saying that you're not happy with deformation groupoids for Galois representations as a natural example of stacks with origins in number theory? –  Keerthi Madapusi Pera May 15 '12 at 3:04
The moduli stack of abelian varieties is certainly more "arithmetic" (also harder (and "more interesting") than M_g, if you think the latter is just motivated by geometry), and it is used in Faltings' proof of the Mordell conjecture. –  temp May 15 '12 at 4:13
@Keerthi - not at all. I'm not an expert in this area, so I don't know what there is out there. I'm proving a result about stacks in general, and having examples where they are used far from my own field is useful. @temp and Keerthi - those are both good answers, if you would like to add them below! –  David Roberts May 15 '12 at 4:30

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.

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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.

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I didn't know the $\Gamma_0(N)$ moduli problem wasn't representable (I don't know yet what it is!), so thanks. –  David Roberts May 15 '12 at 4:32
A more modern reference would be Kai-Wen Lan's thesis (web.math.princeton.edu/~klan/academic.html) where he deals with the arithmetic compactification of PEL Shimura varieties. Or Brian Conrad's Arithmetic moduli of generalized elliptic curves (math.stanford.edu/~conrad/papers/kmpaper.pdf) for a modern source on the special case of modular curves. –  Rob Harron May 15 '12 at 6:20
Thanks, Rob, for the references. –  David Roberts May 16 '12 at 2:52

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)

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Thanks. I took the liberty of hyperlinking the urls you provided. –  David Roberts May 15 '12 at 4:31