# Shtukas for $\mathrm{Spec}\,\mathbf{Z}$

This is a very soft and speculative question. Please feel free to downvote, close or delete it.

Studying the cohomology of moduli spaces of shtukas, Drinfeld proved the Langlands program for $$\mathrm{GL}_2$$ for global function fields, which was later extended by L. Lafforgue to $$\mathrm{GL}_n$$.

Peter Scholze has a conjectural section Shtukas for $$\mathrm{Spec}\,\mathbf{Z}$$ in his 2018 ICM report https://arxiv.org/abs/1712.03708 on arXiv.

Might this be used to attack the Langlands conjectures for $$\mathbf{Q}$$ (and how)?

What could this be used for?

• As you note, such section is vastly speculative, likewise as in a rather large number of other instances in the literature. The truth is that we don't even know how to precisely state the Langlands conjectures over number fields, which is why they are known under the name of "Langlands philosophy". Scholze writes, in that section, that so far his goal is to understand if one can come up with a "universal" cohomology theory (in the spirit of q-de Rham cohomology, for instance) that will be workable with, possibly conjecturally related to motivic cohomology, but concrete – user87684 Jan 11 '18 at 7:22
• and constructed unconditionally on the (all widely open) conjectures on algebraic cycles. Shtukas and cohomology theories appear to be related, an observation that goes back to Drinfeld himself, and the hope is to construct a cohomology theory for algebraic varieties over number fields/number rings, and hopefully prove it is "universal" in an appropriate sense – user87684 Jan 11 '18 at 7:26
• It is highly unlikely that knowledge of such potential concrete construction of such cohomology theory will shed any light whatsoever on the conjectures on algebraic cycles. Albeit such construction ought to be concrete and workable (again, think about q-de Rham cohomology and its site-theoretic version, much alike crystalline cohomology) at some point to do anything really serious with it one will be bound to face an appropriate version of the standard conjectures for it all over again, ie. one should expect a version of the Tate conjecture for such cohomology, and it won't be any easier :) – user87684 Jan 11 '18 at 7:29
• I believe, to sum up, that the point of that section consists in pointing out that it might be possible to construct a cohomology theory for arithmetic schemes, that should morally be motivic in nature, but concrete and workable, in the meantime that all the conjectures on algebraic cycles remain untouched – user87684 Jan 11 '18 at 7:31
• I don't think this question is meaningful/useful to pose here. I'm going to vote to close. – user95222 Jan 11 '18 at 7:35

Scholze's "mixed characteristic shtukas" are already the basis for his attack on the local Langlands conjectures, so in some sense they have already found application to Langlands over $\mathbb{Q}$''. The basic strategy here is to transpose the ideas of Vincent Lafforgue's work on the global Langlands correspondence to shtukas over the Fargues-Fontaine curve.

As for the global problem, it should be noted that Drinfeld's moduli spaces of shtukas are themselves function field analogues of objects that you have (sometimes) in the number field case, namely Shimura varieties. The application to Langlands comes through the fact that the cohomology of moduli spaces of shtukas "realize" the Langlands correspondence. The analogous hope for Shimura varieties is false; for instance we have Shimura varieties for $GL_2$ (i.e. modular curves) but their cohomology cannot capture all automorphic representations, e.g. those attached to Maass forms.

The "universal cohomology theory over $\mathbb{Z}$" doesn't necessarily seem to solve the problem of being able to find all the expected automorphic/Galois representations in geometry, which is the fundamental premise for the work of Drinfeld/Lafforgue, but maybe Scholze has a grand plan for this.

• Just expanding on what you wrote - in the Drinfeld/Lafforgue proof, the Galois-to-automorphic direction, is not proven using shtukas, but rather using the converse theorem and the constructions of L-functions via etale cohomology by Grothendieck et. al. (some cases can instead be done by geometric Langlands theory). This direction remains very hard over number fields, and even a perfect analogue of shtukas wouldn't help much. – Will Sawin Jan 11 '18 at 7:59

After their work on integral $p$-adic Hodge Theory, Bhatt, Morrow and Scholze realized a certain $q$-de Rham cohomology theory was underlying their $A_{\rm inf}$ cohomology in their paper. This seemed to have the nice feature of "interpolating" among all sorts of cohomology theories: $\ell$-adic étale, crystalline, singular.

He's just speculating about how to make this completely precise, and possibly show such cohomology theory, once constructed one day, is "universal". It is unlikely that this will have applications towards global Langlands over number fields, let alone conjectures on algebraic cycles.

It simply seems to me Scholze just wants to understand "what's going on".