Fargues-Fontaine curve

Let $E$ be a nonarchimedean local field with residue field $\mathbb F_q$ , and let $S$ be a perfectoid space over $\mathbb F_q$ . The relative Fargues–Fontaine curve $X_{S,E}$ is the quotient

X_{S,E}=Y_{S,E}/\varphi^{\mathbb Z},

where $Y_{S,E}$ is a punctured adic space built from relative ramified Witt vectors of $S$ , and $\varphi$ is the Frobenius automorphism. The Frobenius action on the relevant punctured locus is free and properly discontinuous.

Geometric case

For $S=\operatorname{Spa}(C^\flat,C^{\flat+})$ , where $C$ is an algebraically closed perfectoid field containing $E$ , one obtains the curve $X_{C^\flat,E}$ . Its algebraic avatar is a regular noetherian one-dimensional scheme. Degree-one closed points encode untilts of $C^\flat$ to characteristic $0$ , together with an embedding of $E$ .

Vector bundles and isocrystals

An isocrystal $(D,\varphi_D)$ over the completed maximal unramified extension of $E$ produces a vector bundle $\mathcal E(D,\varphi_D)$ on the curve. Over an algebraically closed perfectoid base, every vector bundle decomposes uniquely into slope bundles

\bigoplus_{\lambda\in\mathbb Q} \mathcal O(\lambda)^{\oplus m_\lambda}.

This is the Fargues–Fontaine classification and is the curve-level analogue of Dieudonné–Manin slope theory.

Langlands role

The stack $\operatorname{Bun}_G$ of $G$ -bundles on relative Fargues–Fontaine curves is the automorphic space in the geometrization of local Langlands. Its geometric points are indexed by the Kottwitz set $B(G)$ . Modifications at untilt divisors define local Hecke correspondences and local shtuka spaces.

References

Laurent Fargues and Jean-Marc Fontaine, Courbes et fibrés vectoriels en théorie de Hodge $p$ -adique, Astérisque 406, 2018. Numdam.
Laurent Fargues and Peter Scholze, “Geometrization of the local Langlands correspondence,” Chapters II–III. arXiv.