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.

  • $\begingroup$ 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? $\endgroup$ May 15 '12 at 3:04
  • $\begingroup$ 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. $\endgroup$
    – temp
    May 15 '12 at 4:13
  • $\begingroup$ @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! $\endgroup$ 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.


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.

  • $\begingroup$ I didn't know the $\Gamma_0(N)$ moduli problem wasn't representable (I don't know yet what it is!), so thanks. $\endgroup$ May 15 '12 at 4:32
  • 1
    $\begingroup$ 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. $\endgroup$
    – Rob Harron
    May 15 '12 at 6:20

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)

  • 1
    $\begingroup$ Thanks. I took the liberty of hyperlinking the urls you provided. $\endgroup$ May 15 '12 at 4:31

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