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A brief update: in his amazing talk yesterday at MSRI (available here), Laurent Fargues explained (building on work of Peter Scholze, see his phenomenal talk two weeks ago here and his historic Berkeley lectures here) a conjectural picture for the local Langlands correspondence as a geometric Langlands correspondence over the "Fargues-Fontaine curve".

[Caveat Emptor: this is way way out of my depth and hopefully will just spur someone knowledgable to comment.] The "curve" is an adic space, and in fact represents a "diamond" as developed in Scholze's Berkeley lectures. This means a functor on perfectoid rings in characteristic p with some nice representability properties. The curve is closely related to the diamond attached to Q_p itself - the functor which attaches to a perfectoid ring the set of its untilts to characteristic zero.

In any case, Fargues defines the stack Bun_G of the curve, which looks very very coarsely like Bun_G of P^1 --- ie it has a (Harder-Narasimhan) stratification where the pieces are classifying spaces of groups, and correspond to isomorphism classes of Kottwitz' G-isocrystals (Frobenius-twisted conjugacy classes in the p-adic group). On the other hand local systems for the dual group on the curve are precisely Langlands parameters. The conjecture is a functor from these local systems to perverse sheaves on Bun_G which are Hecke eigensheaves --- this notion makes sense thanks to Scholze's new fangled version of the [Beilinson-Drinfeld] affine Grassmannian over Q_p. The functor is required to satisfy other natural analogs of compatibilities from the geometric Langlands program. Fargues ends the talk with various implications of the conjecture, as well as the assertion that it works in the abelian case - ie one should have a purely geometric construction of local class field theory.

Edit: It's important to note that perverse sheaves on Bun_G of the Fargues-Fontaine curve are closely related to representations of p-adic groups. Namely the semistable locus of Bun_G (if I understood correctly) is (very roughly) a union of classifying spaces of groups, which are pure inner forms of our group G over our p-adic field. Hence restricting perverse sheaves to these substacks we obtain a functor to representations. Conversely we might dream of extending sheaves on theses substacks to Hecke eigensheaves on all of Bun_G, and hence attaching to them Langlands parameters. (This might be completely misguided -- but analogous structures do indeed appear in the case of real groups.)

A brief update: in his amazing talk yesterday at MSRI (available here), Laurent Fargues explained (building on work of Peter Scholze, see his phenomenal talk two weeks ago here and his historic Berkeley lectures here) a conjectural picture for the local Langlands correspondence as a geometric Langlands correspondence over the "Fargues-Fontaine curve".

[Caveat Emptor: this is way way out of my depth and hopefully will just spur someone knowledgable to comment.] The "curve" is an adic space, and in fact represents a "diamond" as developed in Scholze's Berkeley lectures. This means a functor on perfectoid rings in characteristic p with some nice representability properties. The curve is closely related to the diamond attached to $\mathbb{Q}_p$ itself - the functor which attaches to a perfectoid ring the set of its untilts to characteristic zero.

In any case, Fargues defines the stack $\mathrm{Bun_G}$ of the curve, which looks very very coarsely like $\mathrm{Bun_G}$ of $\mathbb{P}^1$ --- ie it has a (Harder-Narasimhan) stratification where the pieces are classifying spaces of groups, and correspond to isomorphism classes of Kottwitz' G-isocrystals (Frobenius-twisted conjugacy classes in the p-adic group). On the other hand local systems for the dual group on the curve are precisely Langlands parameters. The conjecture is a functor from these local systems to perverse sheaves on $\mathrm{Bun_G}$ which are Hecke eigensheaves --- this notion makes sense thanks to Scholze's new fangled version of the [Beilinson-Drinfeld] affine Grassmannian over $\mathbb{Q}_p$. The functor is required to satisfy other natural analogs of compatibilities from the geometric Langlands program. Fargues ends the talk with various implications of the conjecture, as well as the assertion that it works in the abelian case - ie one should have a purely geometric construction of local class field theory.

Edit: It's important to note that perverse sheaves on $\mathrm{Bun_G}$ of the Fargues-Fontaine curve are closely related to representations of p-adic groups. Namely the semistable locus of $\mathrm{Bun_G}$ (if I understood correctly) is (very roughly) a union of classifying spaces of groups, which are pure inner forms of our group $\mathrm{G}$ over our p-adic field. Hence restricting perverse sheaves to these substacks we obtain a functor to representations. Conversely we might dream of extending sheaves on theses substacks to Hecke eigensheaves on all of $\mathrm{Bun_G}$, and hence attaching to them Langlands parameters. (This might be completely misguided -- but analogous structures do indeed appear in the case of real groups.)

A brief update: in his amazing talk yesterday at MSRI (available here), Laurent Fargues explained (building on work of Peter Scholze, see his phenomenal talk two weeks ago here and his historic Berkeley lectures here) a conjectural picture for the local Langlands correspondence as a geometric Langlands correspondence over the "Fargues-Fontaine curve".

[Caveat Emptor: this is way way out of my depth and hopefully will just spur someone knowledgable to comment.] The "curve" is an adic space, and in fact represents a "diamond" as developed in Scholze's Berkeley lectures. This means a functor on perfectoid rings in characteristic p with some nice representability properties. The curve is closely related to the diamond attached to Q_p itself - the functor which attaches to a perfectoid ring the set of its untilts to characteristic zero.

In any case, Fargues defines the stack Bun_G of the curve, which looks very very coarsely like Bun_G of P^1 --- ie it has a (Harder-Narasimhan) stratification where the pieces are classifying spaces of groups, and correspond to isomorphism classes of Kottwitz' G-isocrystals (Frobenius-twisted conjugacy classes in the p-adic group). On the other hand local systems for the dual group on the curve are precisely Langlands parameters. The conjecture is a functor from these local systems to perverse sheaves on Bun_G which are Hecke eigensheaves --- this notion makes sense thanks to Scholze's new fangled version of the [Beilinson-Drinfeld] affine Grassmannian over Q_p. The functor is required to satisfy other natural analogs of compatibilities from the geometric Langlands program. Fargues ends the talk with various implications of the conjecture, as well as the assertion that it works in the abelian case - ie one should have a purely geometric construction of local class field theory.

A brief update: in his amazing talk yesterday at MSRI (available here), Laurent Fargues explained (building on work of Peter Scholze, see his phenomenal talk two weeks ago here and his historic Berkeley lectures here) a conjectural picture for the local Langlands correspondence as a geometric Langlands correspondence over the "Fargues-Fontaine curve".

[Caveat Emptor: this is way way out of my depth and hopefully will just spur someone knowledgable to comment.] The "curve" is an adic space, and in fact represents a "diamond" as developed in Scholze's Berkeley lectures. This means a functor on perfectoid rings in characteristic p with some nice representability properties. The curve is closely related to the diamond attached to Q_p itself - the functor which attaches to a perfectoid ring the set of its untilts to characteristic zero.

In any case, Fargues defines the stack Bun_G of the curve, which looks very very coarsely like Bun_G of P^1 --- ie it has a (Harder-Narasimhan) stratification where the pieces are classifying spaces of groups, and correspond to isomorphism classes of Kottwitz' G-isocrystals (Frobenius-twisted conjugacy classes in the p-adic group). On the other hand local systems for the dual group on the curve are precisely Langlands parameters. The conjecture is a functor from these local systems to perverse sheaves on Bun_G which are Hecke eigensheaves --- this notion makes sense thanks to Scholze's new fangled version of the [Beilinson-Drinfeld] affine Grassmannian over Q_p. The functor is required to satisfy other natural analogs of compatibilities from the geometric Langlands program. Fargues ends the talk with various implications of the conjecture, as well as the assertion that it works in the abelian case - ie one should have a purely geometric construction of local class field theory.

Edit: It's important to note that perverse sheaves on Bun_G of the Fargues-Fontaine curve are closely related to representations of p-adic groups. Namely the semistable locus of Bun_G (if I understood correctly) is (very roughly) a union of classifying spaces of groups, which are pure inner forms of our group G over our p-adic field. Hence restricting perverse sheaves to these substacks we obtain a functor to representations. Conversely we might dream of extending sheaves on theses substacks to Hecke eigensheaves on all of Bun_G, and hence attaching to them Langlands parameters. (This might be completely misguided -- but analogous structures do indeed appear in the case of real groups.)

A brief update: in his amazing talk yesterday at MSRI, Laurent Fargues explained (building on work of Peter Scholze) a conjectural picture for the local Langlands correspondence as a geometric Langlands correspondence over the "Fargues-Fontaine curve".

[Caveat Emptor: this is way way out of my depth and hopefully will just spur someone knowledgable to comment.] The "curve" is an adic space, and in fact represents a "diamond" as developed in Scholze's Berkeley lectures. This means a functor on perfectoid rings in characteristic p with some nice representability properties. The curve is closely related to the diamond attached to Q_p itself - the functor which attaches to a perfectoid ring the set of its untilts to characteristic zero.

In any case, Fargues defines the stack Bun_G of the curve, which looks very very coarsely like Bun_G of P^1 --- ie it has a (Harder-Narasimhan) stratification where the pieces are classifying spaces of groups, and correspond to isomorphism classes of Kottwitz' G-isocrystals (Frobenius-twisted conjugacy classes in the p-adic group). On the other hand local systems for the dual group on the curve are precisely Langlands parameters. The conjecture is a functor from these local systems to perverse sheaves on Bun_G which are Hecke eigensheaves --- this notion makes sense thanks to Scholze's new fangled version of the [Beilinson-Drinfeld] affine Grassmannian over Q_p, see Scholze's amazing talk at the Geometric Rep Theory workshop two weeks ago. The functor is required to satisfy other natural analogs of compatibilities from the geometric Langlands program. Fargues ends the talk with various implications of the conjecture, as well as the assertion that it works in the abelian case - ie one should have a purely geometric construction of local class field theory.

A brief update: in his amazing talk yesterday at MSRI (available here), Laurent Fargues explained (building on work of Peter Scholze, see his phenomenal talk two weeks ago here and his historic Berkeley lectures here) a conjectural picture for the local Langlands correspondence as a geometric Langlands correspondence over the "Fargues-Fontaine curve".

[Caveat Emptor: this is way way out of my depth and hopefully will just spur someone knowledgable to comment.] The "curve" is an adic space, and in fact represents a "diamond" as developed in Scholze's Berkeley lectures. This means a functor on perfectoid rings in characteristic p with some nice representability properties. The curve is closely related to the diamond attached to Q_p itself - the functor which attaches to a perfectoid ring the set of its untilts to characteristic zero.

In any case, Fargues defines the stack Bun_G of the curve, which looks very very coarsely like Bun_G of P^1 --- ie it has a (Harder-Narasimhan) stratification where the pieces are classifying spaces of groups, and correspond to isomorphism classes of Kottwitz' G-isocrystals (Frobenius-twisted conjugacy classes in the p-adic group). On the other hand local systems for the dual group on the curve are precisely Langlands parameters. The conjecture is a functor from these local systems to perverse sheaves on Bun_G which are Hecke eigensheaves --- this notion makes sense thanks to Scholze's new fangled version of the [Beilinson-Drinfeld] affine Grassmannian over Q_p. The functor is required to satisfy other natural analogs of compatibilities from the geometric Langlands program. Fargues ends the talk with various implications of the conjecture, as well as the assertion that it works in the abelian case - ie one should have a purely geometric construction of local class field theory.