## Extending finite etale coverings

Fix a nonarchimedean field $K$ of residue characteristic $p$, and let $X$ be a normal rigid analytic space over $K$.  Suppose we’re given a closed nowhere-dense analytic subspace $Z \subset X$ and a finite etale cover $Y$ of $X \smallsetminus Z$.  It’s natural to ask if $Y$ can be extended to a finite cover $Y' \to X$, and whether some further conditions on $Y'$ pin such an extension down uniquely.  Although the analogous problem for complex analytic spaces was solved by Stein and Grauert-Remmert in the 50s (cf. Grauert and Remmert’s article here), there isn’t very much literature on this problem in the rigid analytic context, with the notable exception of Lutkebohmert’s paper, about which more in a minute.  Anyway, it turns out that at least for a base field $K$ of characteristic zero, this problem has a very satisfying answer, and the proof is a fun exercise in swinging lots of big hammers.

First, here’s the precise definition of “cover” which we’ll use.

Definition. Let $X$ be a normal rigid analytic space.  A cover of $X$ is a finite surjective map $\pi: Y \to X$ from a normal rigid space $Y$, such that one of the following two equivalent conditions holds:
1. There exists a closed nowhere-dense analytic subset $Z \subset X$ such that $\pi^{-1}(Z)$ is nowhere-dense and $Y \smallsetminus \pi^{-1}(Z) \to X \smallsetminus Z$ is finite etale.
2.  Each irreducible component $Y_i$ of $Y$ maps surjectively onto an irreducible component $X_i$ of $X$, and contains a point $y_i$ such that $\mathcal{O}_{X,\pi(y_i)} \to \mathcal{O}_{Y_i,y_i}$ is etale.

Equivalence of these conditions is a fun exercise left to the reader; note that the second requirement in 2. is automatic when $K$ has characteristic 0.

Theorem. Let $X$ be a normal rigid analytic space over a characteristic zero nonarchimedean field $K$, and let $Z \subset X$ be any closed nowhere-dense analytic subset.  Then any finite etale cover of $X \smallsetminus Z$ extends uniquely to a cover of $X$.

In other words, the restriction functor from {covers of $X$ etale over $X \smallsetminus Z$} to {finite etale covers of $X \smallsetminus Z$} is an equivalence of categories.

The uniqueness holds without any condition on $K$, and is an easy consequence of a powerful theorem due to Bartenwerfer.  To explain this result, let $X$ be a normal rigid space and let $Z \subset X$ be any closed nowhere-dense analytic subset. Then Barternwerfer proved that any bounded function on $X \smallsetminus Z$ extends (uniquely) to a function on $X$. In particular, if $Y \to X$ is a cover and $U \subset X$ is any open affinoid subset, then $\mathcal{O}_Y(\pi^{-1}(U)) \cong \mathcal{O}_{Y}^{+}(\pi^{-1}(U \smallsetminus U \cap Z))[1/ \varpi]$ depends only on the restriction of $Y$ to $X \smallsetminus Z$. Since the affinoids $\pi^{-1}(U)$ cover $Y$, this gives the desired uniqueness.  More generally, this argument shows that for any closed nowhere-dense analytic subset $Z \subset X$, the restriction functor from covers of $X$ to covers of $X \smallsetminus Z$ is fully faithful.

The existence of an extension is harder, of course.  Until further notice, assume $K$ has characteristic zero.  Note that by the uniqueness argument, we can always work locally on $X$ when extending a finite etale cover of $X \smallsetminus Z$.  Now the key input is the following base case, due to Lutkebohmert:

Theorem (Lutkebohmert): If $X$ is a smooth rigid space and $Z \subset X$ is a simple normal crossings divisor, then any finite etale cover of $X \smallsetminus Z$ extends to a cover of $X$.

This is more or less an immediate consequence of Lemma 3.3 in Lutkebohmert’s paper, although he doesn’t state this result so explicitly (and curiously, he never discusses the uniqueness of extensions).  The main ingredient aside from this Lemma is a result of Kiehl on “tubular neighborhoods”, which says (among other things) that if $D \subset X$ is a snc divisor in a smooth rigid space, then for any point $x$ in $D$ at which $r$ components of $D$ meet, we can find some small affinoid neighborhood $U$ of $x$ in $X$ together with a smooth affinoid $S$ and an isomorphism $U \simeq S \times B^r$ (where $B^r = \mathrm{Sp}K \left\langle X_1, \dots, X_r \right\rangle$ denotes the r-dimensional closed ball) under which the individual components of $D$ meeting $x$ identify with the zero loci of the coordinate functions $X_i$.

Granted these results, we argue as follows.  Clearly we can assume that $X$ is quasicompact.  We now argue by induction on the maximal number $i(D)$ of irreducible components of $D$ passing through any individual point of $X$. Let me sketch the induction informally. If $i(D)=1$, then $D$ is smooth, so Kiehl’s result puts us exactly in the situation covered by the case $r=1$ of Lemma 3.3. If $i(D)=2$, then locally on $X$ we can assume that $D$ has two smooth components $D_1$ and $D_2$. By the previous case, any finite etale cover $Y$ of $X \smallsetminus D$ extends uniquely to covers $Y_i$ of $X \smallsetminus D_i$, which then glue to a cover $Y_0$ of $X \smallsetminus D_1 \cap D_2$.  But now locally along $D_1 \cap D_2$, Kiehl’s result puts is in the situation covered by the case $r=2$ of Lemma 3.3, and then $Y_0$ extends to a cover of $X$.  If $i(D)=3$, then locally on $X$ we can assume that $D$ has three smooth components $D_1, D_2, D_3$. By the previous case, any finite etale cover of $X \smallsetminus D$ extends to a cover $Y_i$ of $X \smallsetminus D_i$, for each $i \in \{1,2,3\}$; here we use the fact that $i(D \smallsetminus D_i) \leq 2$ for $D \smallsetminus D_i$ viewed as a strict normal crossings divisor in $X \smallsetminus D_i$.  Again the $Y_i$‘s glue to a cover $Y_0$ of $X \smallsetminus D_1 \cap D_2 \cap D_3$, and again locally along $D_1 \cap D_2 \cap D_3$ Kiehl’s result puts us in the situation handled by Lemma 3.3, so $Y_0$ extends to a cover of $X$.  Etc.

To get existence in the general case, we use some recent results of Temkin on resolution of singularities.  More precisely, suppose $X = \mathrm{Sp}(A)$ is an affinoid rigid space, and $Z \subset X$ is a closed nowhere-dense subset as before; note that $Z=\mathrm{Sp}(B)$ is also affinoid, so we get a corresponding closed immersion of schemes $\mathcal{Z} = \mathrm{Spec}(B) \to \mathcal{X} = \mathrm{Spec}(A)$.  These are quasi-excellent schemes over $\mathbf{Q}$, so according to Theorem 1.1.11 in Temkin’s paper, we can find a projective birational morphism $f: \mathcal{X}' \to \mathcal{X}$ such that $\mathcal{X}'$ is regular and $(\mathcal{X}' \times_{\mathcal{X}} \mathcal{Z})^{\mathrm{red}}$ is a strict normal crossings divisor, and such that $f$ is an isomorphism away from $\mathcal{Z} \cup \mathcal{X}^{\mathrm{sing}}$.  Analytifying, we get a proper morphism of rigid spaces $g: X' \to X$ with $X'$ smooth such that $g^{-1}(Z)^{\mathrm{red}}$ is an snc divisor etc.

Suppose now that we’re given a finite etale cover $Y$ of $X \smallsetminus Z$.   Pulling back along $g$, we get a finite etale cover of $X' \smallsetminus g^{-1}(Z)$, which then extends to a cover $h: Y'\to X'$ by our previous arguments. Now, since $g \circ h$ is proper, the sheaf $(g \circ h)_{\ast} \mathcal{O}_{Y'}$ defines a sheaf of coherent $\mathcal{O}_X$-algebras. Taking the normalization of the affinoid space associated with the global sections of this sheaf, we get a normal affinoid $Y''$ together with a finite map $Y'' \to X$ and a canonical isomorphism $Y''|_{(X \smallsetminus Z)^{\mathrm{sm}}} \cong Y|_{(X \smallsetminus Z)^{\mathrm{sm}}}$. The cover we seek can then be defined, finally, as the Zariski closure $Y'''$ of $Y''|_{(X \smallsetminus Z)^{\mathrm{sm}}}$ in $Y''$: this is just a union of irreducible components of $Y''$, so it’s still normal, and it’s easy to check that $Y'''$ satisfies condition 1. in the definition of a cover. Finally, since $Y'''$ and $Y$ are canonically isomorphic after restriction to $(X \smallsetminus Z)^{\mathrm{sm}}$, the uniqueness argument shows that this isomorphism extends to an isomorphism $Y'''|_{X \smallsetminus Z} \cong Y$. This concludes the proof.

Combining this existence theorem with classical Zariski-Nagata purity, one gets a purity theorem for rigid spaces:

Corollary. Let X be a smooth rigid analytic space over a characteristic zero nonarchimedean field, and let $Z \subset X$ be any closed analytic subset which is everywhere of codimension $\geq 2$.  Then finite etale covers of $X$ are equivalent to finite etale covers of $X \smallsetminus Z$.

Presumably this result has other fun corollaries.  I’d be happy to know more.

## Diamonds for all!

Regular readers of this blog probably know that I’m obsessed with diamonds.  They can thus imagine my happiness when Peter posted an official foundational reference for diamonds a few weeks ago.

I want to use this occasion to make a remark aimed at graduate students etc. who might be wondering whether they should bother learning this stuff: in my opinion, spending time with difficult* manuscripts like the one above usually pays off in the long run. Of course, this only works if you invest a reasonable amount of time, and there’s some initial period where you’re completely befuddled, but after some months the befuddlement metamorphoses into understanding, and then you have a new set of tools in your toolkit! This shouldn’t be so surprising, though; after all, papers like this are difficult precisely because they are so rich in new ideas and tools.

Really, I’ve had this experience many times now – with the paper linked above and its precedent, with the Kedlaya-Liu “Relative p-adic Hodge theory” series, with Kato’s paper on p-adic Hodge theory and zeta functions of modular forms, etc. – and it was the same every time: for some period of months (or years) I would just read the thing for its own sake, but then at some point something in it would congeal with the rest of the swirling fragments in my head and stimulate me to an idea which never would’ve occurred to me otherwise. It’s the most fun thing in the world. Try it yourself.

*Here by “difficult” I don’t mean anything negative, but rather some combination of dense/forbidding/technical – something with a learning curve.  Of course, there are plenty of papers which are difficult for bad reasons, e.g. because they’re poorly written.  Don’t read them.

## Report from Tucson

Just back from the 2017 Arizona Winter School on perfectoid spaces.  First of all, I should say that everything was impressively well-organized, and that the lecturers did a fantastic job, especially considering the technical weight of this material. (Watch the videos if you don’t believe me.)  Jared Weinstein, in particular, has an almost supernatural ability to make a lecture on some technical thing feel comforting.

Now to the jokes.

• In his opening lecture, Scholze called perfectoid spaces a “failed theory”, on account of his inability to completely settle weight-monodromy. “You see, I’m Prussian, and when a Prussian says he wants to do something, he really feels responsible for doing it.”
• Audience member: “Why are they called diamonds?”
Scholze: “[oral explanation of the picture on p. 63 of the Berkeley notes]”
Weinstein: “Also, diamonds are hard.”
• Anon.: “When you’re organizing a conference, the important thing is not to give in and be the first one who actually does stuff.  Because then you’ll end up doing everything!  Don’t do that!  Don’t be the dumb one!”
Me: “Didn’t you organize [redacted] a couple of years ago?”
Anon.: “Yeah… It turned out that Guido Kings was the dumb one.”
• Mazur: “It just feels like the foundations of this area aren’t yet… hmm…”
Me: “Definitive?”
Mazur: “Yes, exactly.  I mean, if Grothendieck were here, he would be screaming.”
• “Do you ever need more than two legs?”
• During the hike, someone sat on a cactus.
• Finally, here is a late night cartoon of what a universal cohomology theory over $\mathbb{Z}$ might look like (no prizes for guessing who drew this):

## Elliptic curves over Q(i) are potentially automorphic

This spectacular theorem was announced by Richard Taylor on Thursday, in a lecture at the joint meetings.  Taylor credited this result and others to Allen-Calegari-Caraiani-Gee-Helm-Le Hung-Newton-Scholze-Taylor-Thorne (!), as an outcome of the (not so) secret mini-conference which took place at the IAS this fall.  The key new input here is work in progress of Caraiani-Scholze on the cohomology of non-compact unitary Shimura varieties, which can be leveraged to check (at least in some cases) the most difficult hypothesis in the Calegari-Geraghty method: local-global compatibility at l=p for torsion classes.

The slides from my talk can be found here. Naturally I managed to say “diamond” a bunch of times.

## Nowhere-vanishing sections of vector bundles

Let $X/k$ be a proper variety over some field, and let $\mathcal{E}$ be a vector bundle on $X$.  The functor of global sections of $\mathcal{E}$, i.e. the functor sending a scheme $f:S \to \mathrm{Spec}\,k$ to the set $H^0(S \times_{k} X, (f \times \mathrm{id})^{\ast} \mathcal{E})$, is (representable by) a nice affine $k$-scheme, namely the scheme $\mathcal{S}(\mathcal{E}) = \mathrm{Spec}(\mathrm{Sym}_k H^0(X,\mathcal{E})^{\vee})$. Let $\mathcal{S}(\mathcal{E})^{\times} \subset \mathcal{S}(\mathcal{E})$ denote the subfunctor corresponding to nowhere-vanishing sections $s \in H^0(S \times_k X, (f \times \mathrm{id})^{\ast} \mathcal{E})$. We’d like this subfunctor to be representable by an open subscheme. How should we prove this?

Let $p: \mathcal{S}(\mathcal{E}) \to \mathrm{Spec}\,k$ be the structure map. The identity map $\mathcal{S}(\mathcal{E}) \to \mathcal{S}(\mathcal{E})$ corresponds to a universal section $s^{\mathrm{univ}} \in H^0(\mathcal{S}(\mathcal{E}) \times_k X, (p \times \mathrm{id})^{\ast}\mathcal{E})$. Let $Z\subset |\mathcal{S}(\mathcal{E}) \times_k X|$ denote the zero locus of $s^{\mathrm{univ}}$. This is a closed subset. But now we observe that the projection $\pi: \mathcal{S}(\mathcal{E}) \times_k X \to \mathcal{S}(\mathcal{E})$ is proper, hence universally closed, and so $|\pi|(Z)$ is a closed subset of $|\mathcal{S}(\mathcal{E})|$.  One then checks directly that $\mathcal{S}(\mathcal{E})^{\times}$ is the open subscheme corresponding to the open subset $|\mathcal{S}(\mathcal{E})| \smallsetminus |\pi|(Z)$, so we win.

I guess this sort of thing is child’s play for an experienced algebraic geometer, and indeed it took Johan about 0.026 seconds to suggest that one should try to argue using the universal section.  I only cared about the above problem, though, as a toy model for the same question in the setting of a vector bundle $\mathcal{E}$ over a relative Fargues-Fontaine curve $\mathcal{X}_S$. In this situation, $\mathcal{S}(\mathcal{E})$ is a diamond over $S^\lozenge$, cf. Theorem 22.5 here, but it turns out the above argument still works after some minor changes.

## Three things I learned from colleagues this semester

1 (from Patrick Allen) Let $F$ be a number field, and let $\pi$ be a cohomological cuspidal automorphic representation of some $\mathrm{GL}_n(\mathbf{A}_F)$.  Suppose that $\rho_\pi : G_{F,S} \to \mathrm{GL}_n(\overline{\mathbf{Q}_p})$ exists and satisfies local-global compatibility at all places, and that $H^1_f(F, \mathrm{ad}\,\rho_\pi) = 0$ as predicted by Bloch-Kato.  Then the following are equivalent:

a) $H^2(G_{F,S}, \mathrm{ad}\,\rho_\pi) = 0$, as predicted by Jannsen’s conjecture;

b) $H^1(G_{F,S}, \mathrm{ad}\,\rho_\pi)$ has the right dimension;

c) The product of restriction maps $H^1_f(F,\mathrm{ad}\,\rho_\pi(1))\to \prod_{v|p} H^1_f(F_v,\mathrm{ad}\,\rho_\pi(1))$ is injective.

The equivalence of a) and b) follows from Tate’s global Euler characteristic formula, but their equivalence with c) was news to me.  The question of whether or not c) holds came up incidentally in my work with Jack on Venkatesh’s conjecture, so it was very pleasing to learn that it follows from Bloch-Kato + Jannsen.

2 (from Keerthi Madapusi Pera) If $\mathbf{G} / \mathbf{Q}_p$ is semisimple and simply connected, and isotropic (i.e. contains some $\mathbf{Q}_p$-split torus), then $\mathbf{G}(\mathbf{Q}_p)$ has no proper finite-index subgroups.

3 (from Stefan Patrikis) Let $\pi$ be as in 1) again. There are two number fields naturally associated with $\pi$ (besides $F$): the field $\mathbf{Q}(\pi)$ generated by its Hecke eigenvalues, and the “reflex field” $E\subseteq F$ of its cohomological weight.  Is there any chance that $E$ is always a subfield of $\mathbf{Q}(\pi)$?, I asked SP.  Yes, said he.

## Artin vanishing is false in rigid geometry

Sorry for the lack of blogging.  It’s been a busy semester.

Let $k$ be an algebraically closed field, and let $X$ be a $d$-dimensional affine variety over $k$.  According to a famous theorem of Artin (Corollaire XIV.3.5 in SGA 4 vol. 3), the etale cohomology groups $H^i_{\mathrm{et}}(X,G)$ vanish for any $i > d$ and any torsion abelian sheaf $G$ on $X_{\mathrm{et}}$. This is a pretty useful result.

It’s natural to ask if there’s an analogous result in rigid geometry.  More precisely, fix a complete algebraically closed extension $k / \mathbf{Q}_p$ and a $d$-dimensional affinoid rigid space $X=\mathrm{Spa}(A,A^\circ)$ over $k$.  Is it true that $H^i_{\mathrm{et}}(X,G)$ vanishes for (say) any $i>d$ and any $\mathbf{Z}/n\mathbf{Z}$-sheaf $G$ on $X_{\mathrm{et}}$ for $n$ prime to $p$?

I spent some time trying to prove this before realizing that it fails quite badly.  Indeed, there are already counterexamples in the case where $X=\mathrm{Spa}(k \langle T_1,\dots,T_d \rangle,k^\circ \langle T_1, \dots, T_d \rangle)$ is the $d$-variable affinoid disk over $k$.  To make a counterexample in this case, let $Y$ be the interior of the (closed, in the adic world) subset of $X$ defined by the inequalities $|T_i| < |p|$ for all $i$; more colloquially, $Y$ is just the adic space associated to the open subdisk of (poly)radius $1/p$. Let $j: Y \to X$ be the natural inclusion.  I claim that $G = j_! \mathbf{Z}/n\mathbf{Z}$ is then a counterexample.  This follows from the fact that $H^i_{\mathrm{et}}(X,G)$ is naturally isomorphic to $H^i_{\mathrm{et},c}(Y,\mathbf{Z}/n\mathbf{Z})$, together with the nonvanishing of the latter group in degree $i = 2d$.

Note that although I formulated this in the language of adic spaces, the sheaf $G$ is overconvergent, and so this example descends to the Berkovich world thanks to the material in Chapter 8 of Huber’s book.

It does seem possible, though, that Artin vanishing might hold in the rigid world if we restrict our attention to sheaves which are Zariski-constructible.  As some (very) weak evidence in this direction, I managed to check that $H^2_{\mathrm{et}}(X,\mathbf{Z}/n \mathbf{Z})$ vanishes for any one-dimensional affinoid rigid space $X$.  (This is presumably well-known to experts.)