nLab Fargues-Fontaine curve

Idea

Let $C$ be an algebraically closed perfectoid field over $\mathbb{F}_{p}$. The Fargues-Fontaine curve $X_{C}$ is a complete algebraic curve whose closed points parametrize the untilts of $C$. Such an untilt may be recovered as the residue field of the corresponding point.

As an adic space

Let $E$ be a finite extension of $\mathbb{Q}_{p}$ with uniformizer $\pi$ and residue field $\kappa$. Let $S=Spa(R,R^{+})$ be a perfectoid space over $\overline{\mathbb{F}}_{q}$ with pseudouniformizer $\varpi$. We take ramified Witt vectors $W_{\mathcal{O}_{E}}(R):=W(R^{+}\otimes_{W(\kappa)}\mathcal{O}_{E})$. We define $Y_{S}$ to be the locus in $Spa(W_{\mathcal{O}_{E}}(R))$ where $\pi$ and $\varpi$ are invertible. We think of $Y_{S}$ as the “fiber product” $Spa(R^{+})\times Spa(\mathcal{O}_{E})$, even though is is not literally possible because we lack a base for the fiber product (we will define this later)!

The (relative) Fargues-Fontaine curve $X_{S}$ is then defined to be $Y_{S}/\phi_{S}^{\mathbb{Z}}$, where $\phi_{S}$ is the Frobenius on $S$.

In the geometrization of the local Langlands correspondence

The Fargues-Fontaine curve finds application in the geometrization of the local Langlands correspondence. Note that the geometric Langlands correspondence is stated for a curve. Naively, one might think that the “curve” for the local Langlands correspondence should be $Spec(E)$, for $E$ a local field. However, the local Langlands correspondence concerns the Weil group instead of the absolute Galois group, one should instead consider $Spec(\breve{E})/\phi_{E}^{\mathbb{Z}}$, where $\breve{E}$ is the maximal unramified extension of $E$ and $\phi_{E}$ is the Frobenius.

To truly obtain a geometrization, however, one needs to “relativize” this, which is analogous to considering the stack $Bun_{G}$ in the geometric Langlands correspondence instead of just its points $Bun_{G}(\mathbb{F}_{q})$. Hence we must be able to take “base change”.

If $E$ is, say $\mathbb{F}_{q}((t))$, then for a perfectoid space $S=Spa(R,R^{+})$ with pseudouniformizer $\varpi$ we may take the fiber product $S\times_{Spa(\mathbb{F}_{q})} Spa(\mathbb{F}_{q}((t)))$. However if $E$ is a finite extension of $\mathbb{Q}_{p}$, there is no analogue of the base $\mathbb{F}_{q}$ over which we take the fiber product.

However, we note that in the case where $E=\mathbb{F}_{q}((t))$, the fiber product $S\times_{Spa(\mathbb{F}_{q})} Spa(\mathbb{F}_{q}((t)))$ is the locus inside $Spa(R^{+})\times_{Spa(\mathbb{F}_{q})} Spa(\mathbb{F}_{q}[[t]])$ where both the uniformizer $\varpi$ of $R$ and the uniformizer $t$ of $\mathbb{F}_{q}$ are both invertible.

Now noting that the mixed characteristic analogue of forming power series is taking Witt vectors (compare, for instance, $\mathbb{F}_{p}[[t]]$ and $W(\mathbb{F}_{p})=\mathbb{Z}_{p}$), we think of the locus $Y_{S}$ defined above as the analogue of our fiber product, $Spa(R^{+})\times Spa(\mathcal{O}_{E})$. Again, since the local Langlands correspondence is phrased in terms of the Weil group, we have to take the quotient by the action Frobenius, and take $X_{S}=Y_{S}/\phi^{\mathbb{Z}}$, which is the Fargues-Fontaine curve.

As a scheme

If $S=Spa(C)$ where $C$ be an algebraically closed perfectoid field over $\mathbb{F}_{p}$, we have a description of the Fargues-Fontaine curve as a scheme. Let $\mathcal{O}_{C}$ be the ring of integers of $C$, and let $W(\mathcal{O}_{C})$ be the Witt vectors of $\mathcal{O}_{C}$. Let $\phi_{C}:C\to C$ be the Frobenius morphism. We define a norm $\vert\cdot\vert_{r}$ on $W(\mathcal{O}_{C})[1/p]$ as follows:

$$\vert \sum_{n\gg -\infty} [a_{n}]p^{n}\vert_{r}=\sup_{n}\vert a_{n}\vert p^{-r n}$$

Let $B_{C}$ be the Frechet completion of $W(\mathcal{O}_{C})[1/p]$ with respect to all the norms $\vert\cdot\vert_{r}$ for every positive $r$.

Then the schematic Fargues-Fontaine curve $X_{C}^{sch}$ is defined to be

$$X_{C}^{sch}=\mathrm{Proj}(\oplus_{n\in\mathbb{N}} B_{C}^{\phi=p^{n}}).$$

Vector Bundles on the Fargues-Fontaine Curve

Vector bundles? on the Fargues-Fontaine curve admit a classification in terms of isocrystals (vector spaces over the field $E$ together with a semilinear action of Frobenius). More generally for a reductive group $G$, there is a concept of $G$-isocrystals which correspond to $G$-bundles on the Fargues-Fontaine curve.

These also in turn correspond to extended pure inner forms of $G$ (which are all the inner forms if $G$ is connected). It is believed that it should not just be $G$, but all the extended pure inner forms of $G$ together, which should appear in the geometric side of the local Langlands correspondence (for instance inner forms have the same Langlands dual group). This is another motivation for the appearance of the Fargues-Fontaine curve in the Fargues-Scholze geometrization of the local Langlands correspondence.

Related concepts

• p-adic Hodge theory

• local Langlands correspondence

References

• Jared Weinstein, The Fundamental Curve of p-adic Hodge Theory, or How to Un-tilt a Tilted Field

• Jared Weinstein, The Fundamental Curve of p-adic Hodge Theory, Part II

• Dennis Gaitsgory, Jacob Lurie, Harvard Seminar on Fargues-Fontaine Curve (Informal Notes)

• Laurent Fargues, Peter Scholze, Geometrization of the local Langlands correspondence (arXiv:2102.13459)

• a version in global analytic geometry is mentioned at the end of http://www-personal.umich.edu/~snkitche/Conference/notes/Kremnitzer-2.pdf + audio

• Federico Bambozzi, Oren Ben-Bassat, Kobi Kremnizer, Analytic geometry over $\mathbb{F}_1$ and the Fargues-Fontaine curve, (arXiv:1711.04885)