nLab Fargues-Fontaine curve




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

As an adic space

Let EE be a finite extension of β„š p\mathbb{Q}_{p} with uniformizer Ο€\pi and residue field ΞΊ\kappa. Let S=Spa(R,R +)S=Spa(R,R^{+}) be a perfectoid space over 𝔽¯ q\overline{\mathbb{F}}_{q} with pseudouniformizer Ο–\varpi. We take ramified Witt vectors W π’ͺ E(R):=W(R +βŠ— W(ΞΊ)π’ͺ E)W_{\mathcal{O}_{E}}(R):=W(R^{+}\otimes_{W(\kappa)}\mathcal{O}_{E}). We define Y SY_{S} to be the locus in Spa(W π’ͺ E(R))Spa(W_{\mathcal{O}_{E}}(R)) where Ο€\pi and Ο–\varpi are invertible. We think of Y SY_{S} as the β€œfiber product” Spa(R +)Γ—Spa(π’ͺ E)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 SX_{S} is then defined to be Y S/Ο• S β„€Y_{S}/\phi_{S}^{\mathbb{Z}}, where Ο• S\phi_{S} is the Frobenius on SS.

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 β€œcurve” for the local Langlands correspondence should be Spec(E)Spec(E), for EE a local field. However, the local Langlands correspondence concerns the Weil group instead of the absolute Galois group, one should instead consider Spec(breveE)/Ο• E β„€Spec(\breve{E})/\phi_{E}^{\mathbb{Z}}, where breveE\breve{E} is the maximal unramified extension of EE and Ο• E\phi_{E} is the Frobenius.

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

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

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

Now noting that the mixed characteristic analogue of forming power series is taking Witt vectors (compare, for instance, 𝔽 p[[t]]\mathbb{F}_{p}[[t]] and W(𝔽 p)=β„€ pW(\mathbb{F}_{p})=\mathbb{Z}_{p}), we think of the locus Y SY_{S} defined above as the analogue of our fiber product, Spa(R +)Γ—Spa(π’ͺ E)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/Ο• β„€X_{S}=Y_{S}/\phi^{\mathbb{Z}}, which is the Fargues-Fontaine curve.

As a scheme

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

|βˆ‘ nβ‰«βˆ’βˆž[a n]p n| r=sup n|a n|p βˆ’rn\vert \sum_{n\gg -\infty} [a_{n}]p^{n}\vert_{r}=\sup_{n}\vert a_{n}\vert p^{-r n}

Let B CB_{C} be the Frechet completion of W(π’ͺ C)[1/p]W(\mathcal{O}_{C})[1/p] with respect to all the norms |β‹…| r\vert\cdot\vert_{r} for every positive rr.

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

X C sch=Proj(βŠ• nβˆˆβ„•B C Ο•=p n).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 EE together with a semilinear action of Frobenius). More generally for a reductive group GG, there is a concept of GG-isocrystals which correspond to GG-bundles on the Fargues-Fontaine curve.

These also in turn correspond to extended pure inner forms of GG (which are all the inner forms if GG is connected). It is believed that it should not just be GG, but all the extended pure inner forms of GG 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.


Last revised on July 4, 2022 at 01:58:03. See the history of this page for a list of all contributions to it.