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

This is a second post from JW, following on from Part I.

The Galois group of \(\mathbb{Q}_p\) as a geometric fundamental group.

In this follow-up post, I’d like to relay something Peter Scholze told me last fall. It concerns the Galois group \(\mathrm{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)\), and how this is isomorphic to the étale fundamental group of some geometric object \(Z\), which is defined over an algebraically closed field. (Of course, \(\mathrm{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)\) is isomorphic to the étale fundamental group of \(\mathrm{Spec}(\mathbb{Q}_p)\), but that’s tautological.) We can even ask that this isomorphism be “natural” in the sense that there is an equivalence of categories between finite étale covers of \(Z\) and finite étale \(\mathbb{Q}_p\)-algebras. This is the sense in which the absolute Galois groups of a perfectoid field \(F\) and its tilt \(F^\flat\) are naturally isomorphic, cf. the comments following my first post. Anyway, one afternoon during his visit to Boston, Scholze told me the following theorem:

Theorem 1. Let \(C\) be a complete algebraically closed valued field containing \(\mathbb{Q}_p\). There exists an “object” \(Z\) defined over \(C\), which has the property that there is an equivalence of categories between finite étale covers of \(Z\) and finite étale \(\mathbb{Q}_p\)-algebras.

(I will explain later what sort of thing \(Z\) actually is–in brief, it is the quotient by \(\mathbb{Q}_p^\times\) of the punctured perfectoid open disc over \(C\).)

Incredulous, I demanded an explanation, which he gave later that evening, at an Indian restaurant in Harvard Square, with Hadi Hedayatzadeh also present. I left this conversation with a giddy feeling that Theorem 1 could bring a lot of clarity to the local Langlands program. Geometry is easier than arithmetic, after all. If you want to classify representations of \(\mathrm{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)\), now all you have to do is classify local systems on \(Z\), which is surely easier. I envisioned the \(p\)-adic local Langlands correspondence for \(GL(n)\) falling away in a tidy puff.

None of this came to pass (yet). Still, Scholze’s theorem and its proof are really elegant stuff. What follows is a motivated exposition of two curry-stained pages of notes from that conversation last fall. In what follows, \(E\) will always refer to a perfectoid field in characteristic \(p\), and \(F\) will always refer to a perfectoid field in characteristic 0, the idea being that \(E=F^{\flat}\) as usual.

In the last post, we considered the tilting process \(F\mapsto F^{\flat}\), which inputs a perfectoid field in characteristic 0 and outputs a perfectoid field in characteristic \(p\). Then there is an equivalence of categories between étale \(F\)-algebras and étale \(F^\flat\)-algebras.

The tilting process works in families as well:

Theorem 2. Let \(X\) be a perfectoid space over a perfectoid field \(F\) in characteristic 0. Then \(X^{\flat}\) is a perfectoid space over \(F^\flat\), whose underlying topological space is homeomorphic to that of \(X\). There is an equivalence of étale sites \(X_{\text{et}}\cong X^{\flat}_{\text{et}}\).

Thus if we have a perfectoid space in characteristic \(p\), any two of its un-tilts have equivalent étale sites (and therefore the same étale fundamental group). This draws our attention to the problem of un-tilting entire perfectoid spaces. Theorem 1 will be proved by un-tilting a certain perfectoid space in two ways: one will involve the mysterious object \(Z\), and the other will involve the Fargues-Fontaine curve.

Adic spaces and perfectoid spaces.

We need a little background on adic spaces and perfectoid spaces. Let me just recall the main gadgets: one starts with a pair \((R,R^+)\) consisting of a topological ring \(R\) and a bounded open subring \(R^+\), such that the topology on \(R\) is induced by an ideal of \(R^+\) (there are other restrictions as well). Then \(X=\mathrm{Spa}(R,R^+)\) is the set of equivalence classes of continuous valuations \(\left\lvert\;\right\rvert\) on \(R\) which satisfy \(\left\lvert{f}\right\rvert\leq 1\) for all \(f\in R^+\). Under the right hypotheses on \((R,R^+)\), \(X\) forms a topological space equipped two sheaves of rings \(\mathcal{O}_X\) and \(\mathcal{O}_X^+\), whose global sections are \(R\) and \(R^+\), respectively. General adic spaces are formed by gluing together affinoid spaces of the form \(\mathrm{Spa}(R,R^+)\).

A basic example is \(\mathrm{Spa}(\mathbb{Z}_p,\mathbb{Z}_p)\), which has two valuations: the one with \(\lvert{p}\rvert=0\) (the special point) and the one with \(\lvert{p}\rvert\neq 0\) (the generic point, which we’ll call \(\eta_{\mathbb{Q}_p}\)). Another is \(\mathrm{Spa}(\mathbb{Q}_p\langle t \rangle,\mathbb{Z}_p\langle t \rangle)\), which is the adic version of the closed unit disc (note that $latex \lvert{t}\vert
\leq 1$ for all valuations \(\lvert{\;}\vert\)). (This is like the Berkovich unit disc but with another class of exotic points added, corresponding to valuations of rank 2.) If that’s the closed disc, what’s the open disc? (It can’t be of the form \(\mathrm{Spa}(R,R^+)\), since affinoids are always compact.) One way to construct it is to glue together an ascending sequence of closed discs, but the most direct way is to start with \(\mathrm{Spa}(\mathbb{Z}_p\llbracket t \rrbracket,\mathbb{Z}_p\llbracket t \rrbracket)\) and to take its fiber over the generic point of \(\mathrm{Spa}(\mathbb{Z}_p,\mathbb{Z}_p)\), meaning the set of continuous valuations on \(\mathbb{Z}_p\llbracket t \rrbracket\) for which \(\lvert{p}\vert\neq 0\). This is the generic fiber of the formal unit disc \(\mathrm{Spf} \ \mathbb{Z}_p\llbracket t \rrbracket\). We will write this as \((\mathrm{Spf} \ \mathbb{Z}_p\llbracket t \rrbracket)_{\eta}\), where \(\eta=\eta_{\mathbb{Q}_p}=\mathrm{Spa}(\mathbb{Q}_p,\mathbb{Z}_p)\) is the generic point of \(\mathrm{Spa}(\mathbb{Z}_p,\mathbb{Z}_p)\).

Now if \(K\) is a perfectoid field, one has the notion of a perfectoid affinoid \(\mathrm{Spa}(R,R^+)\) over \(K\): this means approximately that \(R\) is a \(K\)-algebra, \(R^+\) is an \(\mathcal{O}_K\)-algebra, and the Frobenius map is surjective on \(R^+/p\). A typical example is \(\mathrm{Spa}(K\langle t^{1/p^{\infty}} \rangle,\mathcal{O}_K\langle t^{1/p^{\infty}} \rangle)\), the perfectoid closed disk. A perfectoid space over \(K\) is an adic spaces admitting a covering by perfectoid affinoids over \(K\). For instance, let

\(D_K=(\mathrm{Spf} \ \mathcal{O}_K\llbracket t^{1/p^\infty} \rrbracket)_{\eta}.\)

I’ll call \(D_K\) the perfectoid open disc over \(K\). The tilting process \(X\mapsto X^{\flat}\) locally looks like \((R,R^+)\mapsto (R^{\flat},R^{\flat,+})\), where \(R^{\flat,+}=\varprojlim R^+/p\) and \(R^\flat=R^{\flat,+}\otimes_{\mathcal{O}_{K^{\flat}}} K^{\flat}\). Then \(D_K^{\flat}=D_{K^{\flat}}\), the perfectoid disc over \(K^\flat\).

Let \(E\) be a perfectoid field in characteristic \(p\), and let \(\varpi\in \mathcal{O}_E\) be any non-unit. Let \(D_E^*\) be the punctured open disc, so that \(D_E^*\) is the set of continuous valuations on \(\mathcal{O}_E\llbracket t^{1/p^\infty} \rrbracket\) for which \(\lvert{t}\rvert\neq 0\) and \(\lvert{\varpi}\rvert\neq 0\).

I am now going to write down two really different un-tilts of \(D_E^*\). One is simply \(D_F^*\), for any un-tilt \(F\) of \(E\). For the other, we notice that \(D_E^*\) isn’t just a perfectoid space over \(E\), it’s also a perfectoid space over the field \(\mathbb{F}_p(\!(t^{1/p^\infty})\!)\). That is, the map

\(\mathbb{F}_p\llbracket t^{1/p^\infty} \rrbracket\to \mathcal{O}_E\llbracket t^{1/p^\infty} \rrbracket\)

induces a morphism

\(\mathrm{Spa}(\mathcal{O}_E\llbracket t^{1/p^\infty} \rrbracket,\mathcal{O}_E\llbracket t^{1/p^\infty} \rrbracket)\to \mathrm{Spa}(\mathbb{F}_p\llbracket t^{1/p^\infty} \rrbracket,\mathbb{F}_p\llbracket t^{1/p^\infty} \rrbracket),\)

in which \(D_E^*\) maps to the generic point. Thus there is a map \(D_E^*\to\mathrm{Spa}(\mathbb{F}_p(\!(t^{1/p^\infty})\!),\mathbb{F}_p\llbracket t^{1/p^\infty} \rrbracket),\) and this presents \(D_E^*\) as a perfectoid space over \(\mathbb{F}_p\llbracket t^{1/p^\infty} \rrbracket\). So, rather like a Necker cube which pops in and out, \(D_E^*\) is simultaneously a perfectoid space over the bases \(E\) and \(\mathbb{F}_p\llbracket t^{1/p^\infty} \rrbracket\).

Let \(K\) be any un-tilt of \(\mathbb{F}_p(\!(t^{1/p^\infty})\!)\). The other un-tilt of \(D_E^*\) will be a perfectoid space \(Y_{E,K}\) over \(K\) whose tilt is \(D_E^*\). Consider the ring \(A= W(\mathcal{O}_E)\hat{\otimes}_{\mathbb{Z}_p} \mathcal{O}_K\). Then we have \(A/p=\mathcal{O}_E\otimes_{\mathbb{F}_p} \mathcal{O}_K/p\), and

$latex
\begin{aligned}
\varprojlim A/p = \ & \mathcal{O}_E\hat{\otimes}_{\mathbb{F}_p} \mathcal{O}_{K^{\flat}} \\
= \ & \mathcal{O}_E\hat{\otimes}_{\mathbb{F}_p} \mathbb{F}_p\llbracket t^{1/p^\infty} \rrbracket \\
=\ & \mathcal{O}_E\llbracket t^{1/p^\infty} \rrbracket.
\end{aligned}
$

This calculation shows that the tilt of \((\mathrm{Spf} \ A)_{\eta_K}\) is the set of continuous valuations on \(\mathcal{O}_E\llbracket t^{1/p^\infty} \rrbracket\) for which \(\lvert{t}\rvert\neq 0\). Let \( Y_{E,K}=(\mathrm{Spf} \ A)_{\eta_K}\backslash\left\{0\right\},\) where \(0\) refers to the valuation on \(A\) pulled back from the valuation on \(W(\mathcal{O}_E/\mathfrak{m}_E)\hat{\otimes}_{\mathbb{Z}_p} \mathcal{O}_K\) (which is an unramified extension of \(\mathcal{O}_K\)). That is, \(Y_{E,K}\) is the set of valuations on \(A\) satisfying \(\lvert{p}\rvert\neq 0\) and \(\lvert{[\varpi]}\rvert\neq 0\). Then the tilt of \(Y_{E,K}\) is the set of valuations on \(\mathcal{O}_E\llbracket t^{1/p^\infty} \rrbracket\) satisfying \(\lvert{t}\rvert\neq 0\) and \(\lvert{\varpi}\rvert\neq 0\), which is exactly \(D_E^*\).

As the notation suggests, \(Y_{E,K}\) is the base change to \(K\) of an adic space \(Y_E\):

\( Y_E=(\mathrm{Spf} \ W(\mathcal{O}_E))_{\eta_{\mathbb{Q}_p}}\backslash\{0\}, \)

this being the set of continuous valuations on \(W(\mathcal{O}_E)\) for which \(\lvert{p}\rvert\neq 0\) and \(\lvert{[\varpi]}\rvert\neq 0\). Note that if \(F\) is an un-tilt of \(E\), we get a valuation on \(W(\mathcal{O}_E)\) by pulling back a valuation on \(\mathcal{O}_F\) through the map \(\theta\colon W(\mathcal{O}_E)\to \mathcal{O}_F\). The valuations arising this way all satisfy \(\lvert{[\varpi]}\rvert\neq 0\). Thus \(Y_E\) contains the set of un-tilts of \(E\), in a ready-made geometric object (an adic space over \(\mathbb{Q}_p\)).

RMB commented on the previous post, asking for an interpretation of the “non-classical” points of \(Y_E\). If \(F\) is any perfectoid field in characteristic 0, then the \(F\)-points of \(Y_E\) correspond to injections \(E\hookrightarrow F^{\flat}\). The “classical” \(F\)-points correspond to injections where \(F^{\flat}/E\) is finite, but one expects there are plenty of points of \(Y_E\) which do not arise this way.

From here it is not difficult to show that:

Proposition 1 The adic space attached to the Fargues-Fontaine curve \(X_E\) is isomorphic to the quotient \(((\mathrm{Spf} \ W(\mathcal{O}_E))_{\eta_{\mathbb{Q}_p}}\backslash\{0\})/\phi_E^{\mathbb{Z}}\).

From now on I’ll use \(X_E\) to denote the adic space (over \(\mathbb{Q}_p\)), rather than the projective curve; then \(X_{E,K}\) is a perfectoid space, equal to \(Y_{E,K}/\phi_E^{\mathbb{Z}}\). We have shown that the tilt of \(Y_{E,K}\) and is equal to \(D_E^*\). Now, the Frobenius map \(\phi_E\colon E\to E\) induces an automorphism of \(Y_E\), which in turn induces an automorphism of \(Y_{E,K}\) and its tilt \(D_E^*\). On the other hand, there is the automorphism \(\phi_t\) of \(\mathcal{O}_E\llbracket t^{1/p^\infty} \rrbracket\), which is \(\mathcal{O}_E\)-linear and sends \(t\) to \(t^p\). The composition of \(\phi_t\) and \(\phi_E\) induces the absolute Frobenius on \(\mathcal{O}_E\llbracket t^{1/p^\infty} \rrbracket\), which induces the identity on \(D_E^*\), since \(\lvert{\;}\vert\) and \(\lvert{\;}\rvert^p\) are equivalent valuations. This shows that the tilt of \(X_{E,K}=Y_{E,K}/\phi_E^\mathbb{Z}\) is equal to \(D_E^*/\phi_t^\mathbb{Z}\). Therefore:

Proposition 2 The étale site of \(D_E^*/\phi_t^\mathbb{Z}\) is equivalent to the étale site of \(X_{E,K}\).

I would now like to specialize a bit. Let \(C\) be an algebraically closed valued field containing \(\mathbb{Q}_p\). The roles of \(F\) and \(E\) will be played by \(C\) and \(C^\flat\), respectively, and we will specialize \(K\) to be the perfectoid field \(\hat{\mathbb{Q}}_p(\mu_{p^\infty})\). The above proposition tells us that the category of finite étale covers of \(D_{C^{\flat}}^*/\phi_t^\mathbb{Z}\) is equivalent to the category of finite étale covers of \(X_{C^{\flat},K}\). At this point we apply a theorem of Fargues-Fontaine:

Theorem 3 After base-changing to an algebraically closed field, \(X_{C^{\flat}}\) is simply connected.

This theorem is a consequence of the classification of vector bundles on \(X_{C^{\flat}}\), which winds up looking a lot like the same classification for the projective line over a field. As a consequence, there is an equivalence of categories \(Y\mapsto H^0(Y,\mathcal{O}_Y)\) between finite étale covers of \(X_{C^{\flat},K}\) and finite étale \(K\)-algebras. Combining this with Prop. 2, we get equivalences between the following:

• Finite etale covers of \(D_{C^{\flat}}^*/\phi_t^\mathbb{Z}\),
• Finite etale covers of \(X_{C^{\flat},K}\),
• Finite etale \(K\)-algebras.

And therefore we get a surprising isomorphism:

\( \pi_1^{\text{et}}(D_{C^\flat}^*/\phi_t^\mathbb{Z}) \cong \mathrm{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p(\mu_{p^\infty})). \)

All we have to do now to prove Thm. 1 is to descend this picture from \(\mathbb{Q}_p(\mu_{p^\infty})\) down to \(\mathbb{Q}_p\). The group on the right has an action of \(\mathrm{Gal}(\mathbb{Q}_p(\mu_{p^\infty})/\mathbb{Q}_p)\cong\mathbb{Z}_p^\times\); this action corresponds to an action of \(\mathbb{Z}_p^\times\) on \(D_{C^{\flat}}^*\), which for \(a\in\mathbb{Z}_p^\times\) is given by the familiar formula \( t\mapsto (1+t)^a-1.\) These actions of \(\mathbb{Z}_p^\times\) and \(\phi_t^\mathbb{Z}\) combine to give an action of \(\mathbb{Q}_p^\times\) on \(D_{C^{\flat}}^*\), in which \(p\) acts by \(t\mapsto t^p\).

In which case, we have an equivalence

• \(\mathbb{Q}_p^\times\)-equivariant finite etale covers of \(D_{C^\flat}^*\),
• Finite etale \(\mathbb{Q}_p\)-algebras.

Lastly, Theorem 1 promised an object \(Z\) over \(C\), not \(C^{\flat}\). But now we can just use the “easy” un-tilt of \(D_{C^\flat}^*\), namely \(D_C^*\), so long as we can check that the action of \(\mathbb{Q}_p^\times\) lifts to \(D_C^*\). It does, and you can even give formulas for the action of an element \(a\in \mathbb{Q}_p^\times\) on \(D_C^*\) (they involve limits, even for \(a=p\)).

(Pedantic note: the right way to view \(D_C\) is that it is the generic fiber of \(\tilde{\mu}_{p^\infty}\), the universal cover of the multiplicative \(p\)-divisible group \(\mu_{p^\infty}\) over the base \(\mathcal{O}_C\). It so happens that \(\tilde{\mu}_{p^\infty}\) is representable by \(\mathrm{Spf} \ \mathcal{O}_C\llbracket t^{1/p^\infty} \rrbracket\)–I explain this in my Arizona Winter School lectures. Taking generic fibers, we find that \(D_C\) is a \(\mathbb{Q}_p\)-vector space object in the category of perfectoid spaces over \(C\), and that \(\mathbb{Q}_p^\times\) acts on \(D_C^*\).)

The tilting equivalence (Thm. 2) now shows that

Theorem 4 There is an equivalence of categories between finite \(\mathbb{Q}_p^\times\)-equivariant étale covers of \(D_C^*\) and finite étale \(\mathbb{Q}_p\)-algebras.

This is the precise form of Thm. 1 that we wanted. For the object \(Z\), we can attempt to take the quotient \(D_C^*/\mathbb{Q}_p^\times\). This quotient looks horrid–we are quotienting by a group action whose orbits are far from being discrete. But, should a reasonable category be found for \(Z\), the étale fundamental group of \(Z\) can only be \(\mathrm{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)\). Brilliant!