Why to believe the Fargues geometrization conjecture?

Question

In the study of the arithmetic local Langlands correspondence, there is a conjecture that was recently (in this decade) formulated by Fargues.

I can't even concisely state the conjecture so I will defer this to the answerers who will presumably do a much better job of it. I can, however, give a link to a paper by Fargues from which, with sufficient effort, you should be able to extract understanding of the conjecture.

Can you show us "a mathematical path" one can walk along to arrive at the conjecture, starting from the facts which you might expect an average third-year graduate student in specializing in arithmetic geometry or automorphic forms to know? A sort of a one-page (or so) explanation of why one should believe the geometrization conjecture. It would be also nice if you explain what is the relation to the geometric Langlands theory as developed by Gaitsgory and others (my guess is that they are only similar in the name but that is just a guess).

The paper of Fargues already provides a 46-page explanation but it would much easier to follow if one has a specific picture in mind to begin with. There is probably more than one answer to this question (all roads lead to Rome!) but there are not that many people in the world who understand the geometrization conjecture so if we get at least one answer that would be fortunate.

Answer 1

We finally have finished our paper, detailing the conjecture! We have also included an extensive introduction that I hope gives some impression of why one might hope for such a statement, and I'll simply refer you there instead of trying to reproduce it here. It also explains that the conjecture is now extremely parallel to the usual story of geometric Langlands, not just in name.

Basically there are three strands converging to this formulation:

-- From the pure representation theory side, the research on the internal structure of $L$-packets, especially in the non-(quasi-)split case, due to Vogan, Kottwitz, and Kaletha, making Kottwitz' set $B(G)$ of $G$-isocrystals appear.

-- From the arithmetic/Shimura variety side, Carayol's conjecture on the realization of the local Langlands correspondence for $\mathrm{GL}_n$ in the Lubin--Tate and Drinfeld tower (proved by Harris--Taylor, Boyer, etc.), its generalization by Kottwitz to the cohomology of Rapoport--Zink spaces (proved in some cases by Fargues), and from there to general local Shimura varieties and even moduli spaces of $p$-adic shtukas.

-- The geometric Langlands program, as interpreted on the Fargues--Fontaine curve.

Another key intermediate step was the analysis of the cohomology of Shimura varieties via pushforward along the Hodge--Tate period map, as in my papers with Caraiani. This perspective is not really mentioned in my work with Fargues, but in his overview paper, Fargues explains some rough local-global-compatibility statements between the conjectural sheaves on $\mathrm{Bun}_G$ and my work with Caraiani.

Personally, the reason I am 100% confident that the conjecture is true is that one could arrive at the conjecture by very naively translating geometric Langlands to the Fargues--Fontaine curve, and that the resulting conjecture is perfectly compatible with everything we expect about $L$-packets, about categorical unramified Langlands (as in recent conjectures and results of Hellmann, Zhu, and Ben-Zvi--Chen--Helm--Nadler), and the realization of local Langlands in the cohomology of moduli spaces of $p$-adic shtukas.

Answer 2

Why believe in it ? Because I said so. There is absolutely no doubt it is true. This is in some sense evident to me now, even more than in 2014 after my talk at the MSRI, in particular after the release of our joint paper with Peter.

As said by Peter. There are numerous evidences coming from everywhere. At some point this is not a coincidence anymore and there is something as I could begin to see in 2014. Here are different steps from my point of view:

• Already in my PHD work, trying to prove some particular cases of this Kottwitz conjecture describing the discrete part of the cohomology of Rapoport-Zink spaces in terms of local Langlands, I could see that Kottwitz sets are very important (see for example my conjecture that says that for each element of $B(G,\mu)$ the associated Newton stratum is non-empty (now proven in a lot of cases))
• When working on those twin-towers thing I tried further to geometrize Jacquet-Langlands. I mean in some sense the isomorphism (with modern notations) $[\Omega^\diamond / \underline{\mathrm{GL}_n (\mathbb{Q}_p)]}\cong [\mathbb{P}^{n-1,\diamond} / \underline{\mathrm{D}^\times}]$ is a geometric form of Jacquet-Langlands. I did not know what to do with this, tried a few things but this was hopeless (forget about this now). Btw I really understood at this point that those Hodge-Tate periods are a big thing, see further the link with modifications of vector bundles on the curve.
• Then there there was this curve thing. It was clear during my work with Fontaine that modifications of vector bundles on the curve are a key thing in the theory. I perfectly remember being super excited when I understood that the Hodge filtration of a filtered $\varphi$-module allows you to modify a vector bundle on the curve (see the talk I gave at Fontaine's conference where I say that we apply a Hecke correspondence (which gave rise to protestations by Fontaine, of course)). It was more and more clear that modifications of vector bundles were a key thing. In particular I really tried to understand this in my article at Laumon's conference: certain modification of vector bundles are the same as local Shtukas !!!! (aka BKF modules). This was later confirmed by the work of Scholze and Weinstein that says that Rapoport-Zink spaces are moduli of modifications of vector bundles. But just pronouncing the word Shtuka is a great indication that we are in a good direction : some Hecke property (aka modifications of vector bundles) is linked to moduli of Shtukas ;)
• There is this property that the curve is geometrically connected and thus its $\pi_1$ is $\mathrm{Gal} ( \overline{\mathbb{Q}}_p | \mathbb{Q}_p)$...this is the starting point I took at the MSRI in 2014 : let's say there is a geometric Langlands, let's writte the Hecke property...you find Kottwitz conjecture describing the discrete part of the cohomology of local Shtuka moduli spaces...
• The work of Kottwitz and Kaletha was fundamental at some point to try to link this to the classical local Langlands as formulated by Kottwitz and Kaletha.
• The work of Caraiani and Scholze is fundamental too to me, I really wanted to understand were does this perverse sheaf come from ? Coupled with the understanding of the cohomology of Igusa varieties (Harris-Taylor, Mantovan, Shin, Caraiani-Scholze)
• One great thing was the classification of G-bundles on the curve and realizing this is Kottwitz set B(G). This is super striking, in particular when you study reduction of G-bundles.

There are plenty of other things that come to my mind. But since the release of my joint paper with Peter I am definitely convinced. In particular the categorical conjecture is more or less evident. The functor \begin{align*} \mathrm{Perf} ( \mathrm{LocSys}_{\hat{G}_{\overline{\mathbb{Q}}_\ell}}) & \longrightarrow D_{lis} ( \mathrm{Bun}_G,\overline{\mathbb{Q}}_\ell ) \\ M & \longmapsto M \ast \mathcal{W}_{\psi} \end{align*} defined by the spectral action can really be understood as some kind of non-abelian Fourier transform. In some sense the Whittaker sheaf is some kind of kernel. Then the geometric conjecture states that this extends (you define the Fourier transform on some subdomain and hope this "extends by continuity") to an equivalence $$ D^b_{coh} ( \mathrm{LocSys}_{\hat{G}_{\overline{\mathbb{Q}}_\ell}}) \xrightarrow{\sim} D_{lis} ( \mathrm{Bun}_G,\overline{\mathbb{Q}}_\ell )^\omega $$ This is really a Fourier type equivalence equivalence like Fourier-Mukai or Fourier-Deligne and so on, outside of the fact that here one side is coherent and the other one is étale (like in the Fourier-Mellin transform for a torus for example). If you put for $M$ a skycraper sheaf on a discrete parameter for $\mathrm{GL}_n$ (an irreducible $Frob$-semi-simple $\ell$-adic representation of $W_E$) this non-abelian Fourier transform is computed locally on a Whittaker stack by Drinfeld ($n=2$) and Laumon more generally. The fact that this is non-zero and gives the good thing should be an analog of Frenkel-Gaitsgory-Vilonen and Gaitsgory in the function field case over a finite field.

Typicaly this geometric conjecture implies that the two possible natural definitions of the stable Bernstein center are the same, see Example X.1.6. This implies this existence of a nice kernel of functoriality too, see Example X.1.7..

Anyway, this conjecture is true, no doubt. I think that one reason why peoples have difficulties to understand why this has to be true is that they are too focused on classical geometric Langlands that is weird thing to me in some sense. But I think at the end it may be possible that the Langlands program has to be retaken from the beginning, typically the good object locally at a place $p$ are not smooth representations $\pi$ of $G(E)$ but complexes $A\in D_{lis} (\mathrm{Bun}_G,\overline{\mathbb{Q}}_\ell)$. As you can see the kernel of functoriality is defined at the level of $D_{lis}$ naturally. It may be possible that in the next years the Langlands program has to be re-though from the beginning, maybe the good objects that can be transferred "naturally" are not automorphic representations but some object in some $D_{lis}(whatever)$. Stay tuned for this.

Answer 3

These notes, from a course Fargues taught at Chicago and transcribed by Sean Howe, are very nice and make a very strong effort to motivate this conjecture and the surrounding theory by analogy with 'honest' holomorphic geometry.

http://www.math.utah.edu/~howe/farguesLL/notes.pdf

Fargues also has given talks about the history of the conjecture, one of which is transcribed here by Tony Feng: https://www.math.mcgill.ca/goren/GEOM_LL.html

I think the discovery of this conjecture was a real surprise! Indeed, Fargues' conjecture says roughly that the (usual, arithmetic) local Langlands correspondence for a reductive group $G$ over $\mathbf{Q}_p$ can be realized as the unramified, geometric Langlands correspondence over the Fargues-Fontaine curve $X$.

I think it's easier to motivate these ideas by starting with Scholze's somewhat different geometrization of the local Langlands program. Namely, in the Berkeley notes, Scholze-Weinstein develop the machinery necessary to construct a version of the moduli stack of shtukas used by V. Lafforgue in his geometric construction of Langlands parameters over function fields:

In Lafforgue's setup, working over a curve $X/\mathbf{F}_q$, you study the moduli stack of shtukas; to an $\mathbf{F}_q$-scheme $S$, shtukas over $S$ are families of $G$-bundles $\mathscr{E}$ over $X \times_{\mathbf{F}_q} S$ equipped with an isomorphism from $\mathscr{E}$ to the Frobenius pullback $(\mathrm{id} \times \mathrm{Fr}_S)^* \mathscr{E}$ on $X \times_{\mathbf{F}_q} S - \{x_1, \ldots, x_n\}$ for $x_1, \ldots, x_n \in X(S)$. Sending $(\mathscr{E}, (x_1, \ldots, x_n))$ to $(x_1, \ldots, x_n)$ gives a map from the moduli stack of these to $X^n$. The cohomology of this moduli stack gives a local system on $X^n$, and studying the collection of these local systems as $n$ varies allows you to define the Langlands parameter. There's also a local version of this construction, replacing $X$ with $\mathbf{D} = \mathrm{Spf } \ \mathbf{F}_q[[t]]$, or $\mathbf{D}^\times = \mathbf{D} - \{0\} = \mathrm{Spa} \ \mathbf{F}_q((t))$.

In the Berkeley lectures, Scholze-Weinstein work out an analogue of this construction in the $p$-adic setting, replacing $X$ with $\mathrm{Spa}\ \mathbf{Z}_p$ or $\mathrm{Spa} \ \mathbf{Q}_p$, which should be thought of as a formal disk (resp. punctured formal disk). To do this, for a characteristic-$p$ perfectoid space $S$, they define an object - a 'diamond' - which can be thought of as "$\mathrm{Spa} \ \mathbf{Z}_p \times_{\mathbf{F}_p} S$". Since $S$ has characteristic $p$, this object has a Frobenius endomorphism, and they can define shtukas with some number $n$ of poles. The stack of these shtukas lives over an object playing the role of "$\mathrm{Spa} \ \mathbf{Z}_p \times_{\mathbf{F}_p} \cdots \times_{\mathbf{F}_p} \mathrm{Spa} \ \mathbf{Z}_p$". Hopefully, these constructions might allow one to run Lafforgue's arguments in this setting, and realize the local Langlands correspondence inside the cohomology of the moduli stack of shtukas. (The major obstacle to doing this currently seems to be the lack of a good theory of perverse sheaves on these objects).

In the course of the Berkeley conference, Fargues worked out a reformulation of the above story using the Fargues-Fontaine curve, yielding his geometrization conjecture. Essentially, the Fargues-Fontaine curve is an honest scheme which models the diamond $\mathrm{Spa}\ \mathbf{Q}_p \times_{\mathbf{F}_p} \mathrm{Spa} \ \mathbf{C}_p^\flat/ (\mathrm{id} \times \mathrm{Frob}_{\mathbf{C}_p^\flat})$, and there is a "relative Fargues-Fontaine curve" $X_S$ for any characteristic $p$ perfectoid space $S$ which models $\mathrm{Spa} \ \mathbf{Q}_p \times_{\mathrm{F}_p} S/ \mathrm{id} \times \mathrm{Frob}_S$. This means that shtukas over $S$ can instead be thought of as modifications of vector bundles on $X_S$. Now, instead of studying the moduli space of shtukas, one can study the moduli space of vector bundles on the Fargues-Fontaine curve. A major advantage of this reformulation is that it gracefully handles ramification, whereas in the shtuka setup, you need to study shtukas on $\mathrm{Spa} \ \mathbf{Q}_p$ with some sort of "level structure" over $\mathbf{Z}_p$. (I'll admit that I don't fully know how to articulate this point...)

Working out what this should say about the Langlands correspondence, one arrives at Fargues' conjecture, which takes a form remarkably similar to the unramified geometric Langlands correspondence over the Fargues-Fontaine curve.

EDIT: (this was too long to be a comment) Thinking about this some more, I should mention another advantage of Fargues' conjecture over an approach based on moduli of $p$-adic shtukas. The $p$-adic shtukas are (morally) families of vector bundles on the "punctured open disk" $(\mathrm{Spa} \mathbf{Q}_p)^\lozenge \rightarrow (\mathrm{Spa} \mathbf{F}_p)^\lozenge$. Since this space is "non-compact" (in contrast to the Lafforgue setup), bundles have lots of sections, maps between them, etc. These types of problems are one reason why geometric Langlands for an actual punctured disk "lives one category level higher" than geom. Langlands for a projective curve. In this language, issues of ramification should be thought of as understanding the singularities at the puncture point.

On the other hand - and this is the source of much of the excitement - Fargues-Fontaine discovered that the category of vector bundles on the Fargues-Fontaine curve $X$ behaves very similarly to the category of vector bundles on $\mathbf{P}^1$ over an algebraically closed field (e.g. vector bundles are determined by their Harder-Narasimhan polygons, and there is a unique stable bundle with any rational slope). This makes the study of the moduli space of $G$-bundles on $X$ more tractable, and geometrically highlights structures that are important in the representation theory of $G(\mathbf{Q}_p)$, as well as its inner forms.

On the "Galois"/"spectral" side, the fundamental group of $X$ is isomorphic to $\mathrm{Gal}(\mathbf{Q}_p)$, so Galois representations are the same thing as unramified local systems on $X$, just as in the geometric Langlands theory over $\mathbf{C}$.