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dorebell
dorebell
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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 $\mathbf{C}$.

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 $\mathbf{C}$.

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dorebell
dorebell
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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.