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Fargues-Scholze famously describe arithmetic local Langlands via global geometric Langlands on the Fargues-Fontaine (FF) curve.

The FF curve acts like an algebraic curve over $\mathbb{C}_p$ (its residue field), while its fundamental group is either the absolute Galois group or Weil group of $\mathbb{Q}_p$, depending on the version, and its ring of global functions is $\mathbb{Q}_p$. Furthermore, the FF curve is closely related to $\operatorname{Spd}{\mathbb{Q}_p}$, defined by Scholze's theory of diamonds.

Do we expect there to one day be an object, let's call it $X$, such that geometric Langlands on $X$ is related to global arithmetic Langlands over $\mathbb{Q}$? One might expect $X$ (or its different versions) to have fundamental group either the absolute Galois group, Weil group, or Langlands group (or beyond) of $\mathbb{Q}$, have ring of global functions $\mathbb{Q}$, and residue fields corresponding to $\mathbb{C}=\mathbb{C}_{\infty}$ and $\mathbb{C}_p$ for different $p$.

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    $\begingroup$ Scholze's talk on $\operatorname{Spec}\mathbb Z$ (mpim-bonn.mpg.de/node/12330) might be closely related, and the course of analytic stacks (youtube.com/playlist?list=PLx5f8IelFRgGmu6gmL-Kf_Rl_6Mm7juZO). $\endgroup$
    – Z. M
    Commented Jun 23 at 15:55
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    $\begingroup$ As Dustin quite explicitly says in the first lecture of the course on analytic stacks (and maybe I also say something like this in my first lecture), my primary motivation for developing this theory of analytic stacks is to make something along those lines work. It will, however, look a bit different from what you describe. My ICM report outlines a little bit of this philosophy of "shtukas over $\mathrm{Spec}(\mathbb Z)$". My current work on real local Langlands gives the piece at the real numbers. $\endgroup$
    – Peter Scholze
    Commented Jun 23 at 20:08
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    $\begingroup$ To expand on one minor aspect of @PeterScholze's "look a bit different": a dream model here is not geometric Langlands on anything (which is an equivalence of categories) but something analogous to global Langlands for function fields: an isomorphism of vector spaces of "functions" on suitable "Bun" and "Loc" in the unramified setting (a la Arinkin et al: AGKRRV) or more generally an identification of objects in categories of local data at ramified places - and it is THOSE ambient local categories which have a geometric Langlands description. $\endgroup$
    – David Ben-Zvi
    Commented Jun 24 at 0:27
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    $\begingroup$ @DavidBen-Zvi Indeed, global Langlands should be about matching objects in (the (restricted) tensor product of) the categories that are identified under local Langlands. There ought to be some global analogue of the Fargues--Fontaine curve, and a moduli space of $G$-bundles on this global object; this should map to the (restricted) product of the moduli spaces of $G$-bundles on the local curves. On the automorphic side, the intended object is the $!$-pushforward of the constant sheaf along this map. On the Galois side, there is a similar picture with spaces of global and local $L$-parameters. $\endgroup$
    – Peter Scholze
    Commented Jun 24 at 8:29

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