Persiflage

Parenthood

Posted on September 9, 2013 by Persiflage

Some questions, I guess, one can’t be prepared for:

$why didn't you?$

Posted in Waffle | Tagged Frankfurt, Poland | Leave a comment

Question: When you are sick in bed, can you do any mathematics? I just spent the past few weeks with a sinus infection and was completely unable to do anything productive, that is, apart from writing an NSF grant (which I guess was somewhat productive). Oh, and I did watch the entire series of “the thick of it” and a bunch of QI episodes.

Some further notes on my trip to Europe. I met some people for the first time. Kathrin Bringmann was very nice; we chatted a little about the possibility of a representation theoretic theory of mock automorphic forms. Harald Helfgott talked about his recent work (nicely described here), but also made constant references during lunch to the consumption of human flesh (perhaps the two are related). He also bested me in a game of chess (my excuse is that there was an implicit agreement to to play as fast as possible, but after a blunder gave him a winning position, he grounded it out like a man who knows he’s a few log factors ahead).

Samir Sisek gave a talk about his nice paper with with Freitas on Fermat over some totally real fields, in which they prove (for example) that there are no solutions for large enough primes $p$ over all real quadratic fields $\mathbf{Q}(\sqrt{d})$ with $d$ squarefree and $3 \mod 8$. I had actually given the problem of proving Fermat for infinitely many real quadratic fields to one of my students a few weeks ago, so this was good timing, I guess.

Bilu, Parent, and Rebolledo gave a great sequence of talks on there work. This was always a paper that I thought I should read at some point, but now I don’t have to. Excellent!

The work of Roberts and Venkatesh is fun, I may blog about that when Roberts gives a talk at Northwestern in a few months time.

Don Zagier has a function which he cannot define, and I have a function which I cannot compute. We conjecture that they are the same. Don also gave me the following puzzle, which I pass on to you:

Show this bold Prussian that brings slaughter, slaughter brings rout!

Posted in Mathematics, Waffle | Tagged Akshay Venkatesh, Bilu, Bringmann, Dave Roberts, Harald Helfgott, Parent, Puzzle, QI, Rebolledo, Samir Siksek, Sickness, Zagier | Leave a comment

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

Posted on August 12, 2013 by Persiflage

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!

Posted in Mathematics, Uncategorized | Tagged Jared Weinstein, Perfectoid Spaces, Peter Scholze, Tilting | Leave a comment

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

Posted on July 23, 2013 by Persiflage

As Quomodocumque once said concerning the most recent set of courses at Arizona Winter School, “Jared Weinstein [gives] a great lecture.” On that note, I am delighted to welcome our first guest post, by the man himself. Note that it has been converted from LaTeX into “wordpress” flavour of LaTeX, so any errors were probably introduced by me in the conversion.

************

In any treatment of $ p$-adic Hodge theory, one inevitably encounters a procedure for passing from fields of characteristic 0 to fields of characteristic $ p$. In modern parlance, if $ F$ is a perfectoid field of characteristic 0, one has the tilt $ F^\flat$, a perfectoid field of characteristic $ p$. (For instance, the construction of Fontaine’s period ring $ B_{\text{dR}}$ has $ \mathbb{C}_p^{\flat}$ as an intermediate step.) What happens when you try to un-tilt? That is, given a perfectoid field $ E$, can we describe the set of all perfectoid fields $ F$ with $ F^{\flat}=E$? This question leads us right into the subject of our post, the remarkable “fundamental curve of $ p$-adic Hodge theory”, due to Fargues and Fontaine. (See here, and also an English summary here.)

First we review the tilting procedure. Start with a field $ F$ of characteristic 0 which is complete with respect to a nonarchimedean absolute value, whose residue field is perfect of characteristic $ p$. (Pedantic note: The absolute value itself doesn’t come packaged with $ F$, only the topology does.) Let $ {\mathcal{O}}_F$ be its ring of integers. Then form

$ {\mathcal{O}}_{F^\flat}= \lim_{x\mapsto x^p} {\mathcal{O}}_F/p. $

Then $ {\mathcal{O}}_{F^\flat}$ is a ring in characteristic $ p$. It’s not hard to see that $ {\mathcal{O}}_{F^\flat}$ is a domain. Let $ F^{\flat}$ be its fraction field.
Typically we only care about the case when $ F$ is a perfectoid field, which means that it satisfies the following two properties:

The value group $ \left|{F^\times}\right|$ is non-discrete.
The Frobenius map $ x\mapsto x^p$ is surjective on $ {\mathcal{O}}_F/p$.

The field $ {\mathbb{Q}}_p$ satisfies the second property but not the first, and $ {\mathbb{Q}}_p^\flat = {\mathbb{F}}_p$. If $ \ell$ is prime to $ p$, then the completion $ F$ of $ {\mathbb{Q}}_p(p^{1/\ell^\infty})$ satisfies the first property but not the second, and $ F^{\flat}={\mathbb{F}}_p$ also. In these examples the passage from $ F$ to $ F^\flat$ is intolerably lossy. But if $ F$ is a perfectoid field, then it turns out that $ F^\flat$ is another perfectoid field.

So suppose $ F$ is perfectoid. To see where the topology on $ F^\flat$ comes from, one has to observe the isomorphism of multiplicative monoids (not rings)

$ F^{\flat}=\lim_{x\mapsto x^p} F. $

(This is a good exercise if you haven’t seen this before. Also, for this it is important that $ F$ be complete.)
Given an element $ e\in F^{\flat}$ which corresponds to $ (f,f^{1/p},\dots)$ in the above bijection, we put $ e^\sharp = f$. If $ \left| \; \right|$ is an absolute value which defines the topology on $ F$, we can define a corresponding absolute value on $ F^\flat$ by setting $ \left|{e}\right|=\left|{e^\sharp}\right|$.

For instance, let $ F=\hat{{\mathbb{Q}}}_p(\mu_{p^\infty})$ be the completion of the field obtained by adjoining all $ p$th power roots of unity to $ {\mathbb{Q}}_p$. Then $ {\mathcal{O}}_{F^{\flat}}$ contains an element $ t=(0,1-\zeta_p,1-\zeta_{p^2},\dots)$ which is topologically nilpotent, as well as a system of roots $ t^{1/p^n}$. This means that $ {\mathcal{O}}_{F^{\flat}}$ must contain the ring $ {\mathbb{F}}_p[[{t^{1/p^\infty}}]]$, this being the $ t$-adic completion of $ {\mathbb{F}}[t^{1/p^\infty}]$. In fact $ F^{\flat}={\mathbb{F}}_p(\!({t^{1/p^\infty}})\!)$, the $ t$-adic completion of $ {\mathbb{F}}_p(t^{1/p^\infty})$. Similarly, if $ F=\hat{{\mathbb{Q}}}_p(p^{1/p^\infty})$, we may set $ t=(0,p^{1/p},p^{1/p^2},\dots)$, and then once again $ F^{\flat}={\mathbb{F}}_p(\!({t^{1/p^\infty}})\!)$.

Probably the most striking relationship between $ F$ and $ F^{\flat}$ is this:

Theorem A: The absolute Galois groups of $ F$ and $ F^{\flat}$ are ~~naturally~~ isomorphic.

The precise statement of this theorem is that there is an equivalence of categories between finite étale $ F$-algebras and finite étale $ F^\flat$-algebras. Applied to $ F=\hat{{\mathbb{Q}}}_p(\mu_{p^\infty})$, Theorem A lies at the heart of the construction of $ (\phi,\Gamma)$-modules attached to $ p$-adic Galois representations. Scholze’s work on perfectoid spaces provides a version of Theorem A that works in families; using this he was able to prove Deligne’s weight-monodromy conjecture for a hypersurface over $ {\mathbb{Q}}_p$ by wrestling it into characteristic $ p$, where the conjecture was known previously.

Anyway, we promised to talk about how to un-tilt. Thus suppose we are given $ E$, a perfectoid field in characteristic $ p$. Does there exist a valued field $ F$ with $ F^{\flat}=E$? If so, is $ F$ unique? Evidently not: We have just seen that the fields $ \hat{{\mathbb{Q}}}_p(\mu_{p^\infty})$ and $ \hat{{\mathbb{Q}}}_p(p^{1/p^\infty})$ both have tilt $ {\mathbb{F}}_p(\!({t^{1/p^\infty}})\!)$. So there are at least two ways to un-tilt the latter field. Let’s make precise what we mean by un-tilt:

Definition: An un-tilt of $ E$ is an isomorphism class of pairs $ (F,\iota)$, where $ F$ is a perfectoid field of characteristic 0 and $ \iota\colon E\hookrightarrow F^{\flat}$ is an embedding of topological fields, such that $ F^{\flat}/\iota(E)$ is a finite extension. (Two such pairs $ (F,\iota)$ and $ (F’,\iota’)$ are isomorphic if there is an isomorphism $ F\cong F’$ making the obvious diagram commute.) The degree of $ (F,\iota)$ is the degree of $ F^{\flat}/\iota(E)$.
Let $ \left|{Y_E}\right|$ be the set of un-tilts of $ E$.

The idea behind these definitions is that there ought to be some kind of geometric object $ Y_E$ (something like a rigid space) whose set of closed points $ \left|{Y_E}\right|$ parametrize un-tilts. I should explain why I’m including un-tilts of degree $ >1$, rather than using a stricter definition requiring that $ \iota$ be an isomorphism. The reason is that if $ E’/E$ is a Galois extension with group $ G$, then we want $ G$-orbits of $ \left|{Y_{E’}}\right|$ to be in bijection with $ \left|{Y_E}\right|$, and the definition is exactly what is necessary to make this happen. Note that a $ G$-orbit of size $ g$ constisting of points of $ \left|{Y_{E’}}\right|$ of degree $ d$ corresponds to a single point of $ \left|{Y_E}\right|$ of degree $ dg$.

Let $ \phi\colon E\to E$ be the $ p$th power Frobenius automorphism. Then there is an action of $ \phi^{\mathbb{Z}}$ on $ \left|{Y_E}\right|$, given by sending $ (F,\iota)$ to $ (F,\iota\circ\phi^n)$. We wanted to parametrize the un-tilts of $ F$, but it seems like two un-tilts which differ by $ \phi^{\mathbb{Z}}$ aren’t all that different. Let us call two un-tilts of $ E$ equivalent if they differ by some power of Frobenius, so that the set of equivalence classes of un-tilts of $ E$ is $ \left|{Y_E}\right|/\phi^{\mathbb{Z}}$.

This is one of the main theorems of Fargues-Fontaine:

Theorem B: There exists a complete$ \dagger$ curve* $ X_E$ whose closed points are naturally in bijection with equivalence classes of un-tilts of $ E$. If $ x\in \left|{X_E}\right|$ corresponds to the class of the un-tilt $ (F,\iota)$, then $ F$ is the residue field of $ x$.

I now have to explain the asterisk and the dagger, and in doing so I will try to get across just how strange the object $ X_E$ is. First, the asterisk. A “curve” is a separated integral noetherian scheme which is regular of dimension 1. In other words, a curve is built by gluing together spectra of Dedekind rings. Thus $ \mathrm{Spec}\ {\mathbb{Z}}$ is a curve, and so are the affine line $ \mathbb{A}^1_K$ and the projective line $ \mathbb{P}^1_K$ over any field $ K$. In the latter two examples, the residue fields of closed points are finite extensions of the base field $ K$. But un-tilts of $ E$ don’t seem to lie over any common base field–recall that the fields $ \hat{{\mathbb{Q}}}_p(\mu_{p^\infty})$ and $ \hat{{\mathbb{Q}}}_p(p^{1/p^\infty})$ are both un-tilts of $ {\mathbb{F}}_p(\!({t^{1/p^\infty}})\!)$, and these fields are certainly not finite over any common subfield.

So perhaps $ X_E$ is more like $ {\mathrm{Spec} \ } {\mathbb{Z}}$, in the sense that it doesn’t admit a finite type morphism to any $ {\mathrm{Spec} \ } K$, for $ K$ a field. Fine, except that $ X_E$ is also complete. What does complete mean here, if not that it admits a proper morphism to some $ {\mathrm{Spec} \ } K$? Fargues and Fontaine define it this way: A complete curve is a curve $ X$ admitting a map $ \deg\colon \left|{X}\right| \to {\mathbb{Z}}$, such that the degree of any principal divisor is 0. Theorem B then says that $ X_E$ is complete with respect to the degree map which has already been defined on un-tilts.

Is there an analogue of Theorem B for $ Y_E$? Not quite. The situation is analogous to the situation of the Tate curve in rigid-analytic geometry. Let $ Y$ be the multiplicative group $ \mathbb{G}_m$, considered as a rigid-analytic space over $ {\mathbb{Q}}_p$, and let $ q\in {\mathbb{Z}}_p$ have positive valuation. Then $ q$ acts discontinously on $ Y$ without fixed points, one can form the quotient $ X=Y/q^{\mathbb{Z}}$, which ends up being the analytification of an elliptic curve with $ j$-invariant $ j(q)$. So $ X$ is a complete curve and $ Y$ is not.

Let’s sketch the construction of $ X_E$. Since $ X_E$ is supposed to parametrize un-tilts of $ E$, which are in characteristic 0, perhaps it is not surprising that Witt vectors get involved. Say $ x\in Y_E$ corresponds to $ (F,\iota)$. Then the sharp map $ \sharp\colon {\mathcal{O}}_E\to{\mathcal{O}}_F$ induces an honest ring homomorphism $ W({\mathcal{O}}_E)\to {\mathcal{O}}_F$, characterized by $ [e]\mapsto e^\sharp$. This extends to a surjective homomorphism $ \theta_x\colon W({\mathcal{O}}_E)[1/p]\to F$, which we also write as $ f\mapsto f(x)$. The kernel of $ \theta_x$ is a maximal ideal, so we get a map $ \left|{Y_E}\right| \to {\mathrm{MaxSpec} \ } W({\mathcal{O}}_E)[1/p]$. Unfortunately I highly doubt this map is a bijection–$ W({\mathcal{O}}_E)[1/p]$ probably has complicated maximal ideals whose residue fields aren’t un-tilts of $ E$.

It seems that $ W({\mathcal{O}}_E)[1/p]$ is not the full ring of functions on $ Y_E$. We’ll construct a ring $ B_E$ which contains $ W({\mathcal{O}}_E)[1/p]$ which has the property that closed maximal ideals of $ B_E$ are in bijection with $ \left|{Y_E}\right|$. The construction is analytic in nature. Let $ \left|{\;}\right|$ be an absolute value on $ E$ which induces its topology. For $ r>0$, define a norm $ \left|{\;}\right|_r$ on $ W({\mathcal{O}}_E)[1/p]$ by

$ \left|{\sum_{n\gg -\infty} [a_n] p^n}\right| = \sup_n \left|{a_n} \right| p^{-rn}. $

Now suppose $ x=(F,\iota)$ is an un-tilt of $ F$. Let $ \left|{\;}\right|_F$ be the absolute value on $ F$ for which $ \left|{p}\right|_F=1/p$. This absolute value induces $ \left|{\;}\right|_{F^{\flat}}$ on $ F^{\flat}$, which is a finite extension of $ E$ via $ \iota$. The two absolute values on $ E$ must be equivalent, in the sense that there exists $ r>0$ for which $ \left|{e}\right|^r=\left|{\iota(e)}\right|_{F^\flat}$ for all $ e\in E$. Thus $ \left|{e}\right|^r=\left|{e^\sharp}\right|_F$.

Given $ f\in W({\mathcal{O}}_E)[1/p]$, we can compare $ \left|{f(x)}\right|_E$ and $ \left|{f}\right|_r$. If $ f=\sum [a_n]p^n$, then

$
\begin{aligned} \left|{f(x)}\right|_F = & \ \left|{\sum a_n^\sharp p^n}\right|_F\\
\leq & \ \sup \left|{a_n^{\sharp}}\right|_Fp^{-n} \\
= & \ \sup \left|{a_n}\right|^{1/r}p^{-n}\\
= & \ \left|{f}\right|_r^{1/r}
\end{aligned}$

It follows from this inequality that if $ f_i$ is a Cauchy sequence in $ W({\mathcal{O}}_E)[1/p]$ with respect to $ \left|{\;}\right|_r$, then $ f_i(x)$ converges in $ F$.

Definition: Let $ B_E$ be the Fréchet completion of $ W({\mathcal{O}}_E)[1/p]$ with respect to the norms $ \left|{\;}\right|_r$ for $ r>0$. That is, $ B_E$ is the ring of sequences in $ W({\mathcal{O}}_E)[1/p]$ which are Cauchy with respect to every $ \left|{\;}\right|_r$, modulo those sequences which converge to 0 with respect to every $ \left|{\:}\right|_r$.

In light of the foregoing discussion, if $ f\in B_E$, then $ f(x)$ makes sense for any $ x\in \left|{Y_E}\right|$. Thus for every un-tilt $ (F,\iota)$ of $ E$, we get a continuous surjection $ B_E\to F$ extending $ \theta_x$, whose kernel is a closed maximal ideal of $ B_E$.

Theorem C: Closed maximal ideals of $ B_E$ are in bijection with $ \left|{Y_E}\right|$.

This tempts us to define $ Y_E$ as a rigid space by setting $ Y_E={\mathrm{MaxSpec} \ } B_E$, except that $ B_E$ isn’t anything like a Tate algebra. It turns out that $ Y_E$ can be given a meaningful definition as an adic space, but this is the topic for another post.

At this point we can link $ Y_E$ to classical $ p$-adic Hodge theory. If $ F$ is a perfectoid field in characteristic 0, and $ E=F^{\flat}$, then we get a point $ \infty\in \left|{Y_E}\right|$, and a maximal ideal $ \mathfrak{m}\subset B_E$. Then the completion of $ B_E$ with respect to $ \mathfrak{m}$ is $ B_{dR,F}^+$, the de Rham period ring associated to $ F$. This is a complete DVR with residue field $ F$.

Now we turn to $ X_E$, which ought to be the quotient $ Y_E/\phi^{\mathbb{Z}}$. This is supposed to be something like a projective curve, and I would like to motivate the construction of $ X_E$ with projective curves in mind. To that end, suppose $ X$ is a curve which is proper over a field, and you would like to give some kind of explicit presentation for $ X$. (For intance, $ X$ could be a smooth curve of genus 1, and you would like to show that $ X$ is isomorphic to a plane cubic.) The usual thing to do is to find a very ample line bundle $ \mathscr{L}$ on $ X$, in which case

$ X={\mathrm{Proj} \ }\left(\bigoplus_{n\geq 0} H^0(X,\mathcal{L}^{\otimes n})\right).$

In the case of the Fargues-Fontaine curve $ X_E$, what should $ \mathscr{L}$ be? Whatever it is, it must pull back to a line bundle on $ Y_E$ which is $ \phi$-equivariant. Since $ B_E$ is the ring of analytic functions on $ Y_E$, this should be the same as giving a free $ B_E$-module $ M$ of rank 1 together with a $ \phi$-semilinear map $ \phi\colon M\to M$. Let $ M=B^+e$, where $ \phi(e)=p^{-1}e$. This corresponds to a line bundle $ \mathscr{L}$ on the (not yet defined) $ X_E$. Then we ought to have, for any $ n\in{\mathbb{Z}}$,

$ H^0(X_E,\mathscr{L}^{\otimes n})=(Be^{\otimes n})^{\phi=1}=B^{\phi=p^n}. $

This prompts the following definition.

Definition: $ X_E={\mathrm{Proj} \ } P$, where $ P=\oplus_{n\geq 0} P_n$ is the graded $ {\mathbb{Q}}_p$-algebra with $ P_n=B_E^{\phi=p^n}$

To convince you this was the right thing to do, let me list the following facts, which hold when $ E$ is algebraically closed:

$ P$ is a graded factorial ring, whose irreducible homogeneous elements are exactly the nonzero elements of degree 1.
If $ t\in P_1$ is nonzero, then its divisor in $ X_E$ is $ (\infty_t)$, for a point $ \infty_t\in \left|{X_E}\right|$ of degree 1.
Conversely, if $ \infty\in \left|{X_E}\right|$ then there exists $ t\in P_1$ whose divisor is $ (\infty_t)$.
More generally, the divisor of a nonzero element of $ P_n$ has degree $ n$.

(I didn’t say exactly what the divisor of an element $ f\in B_E$ is, but it’s what you think: a formal sum of points in $ \left|{X_E}\right|$, weighted with multiplicities. Since each $ B_{dR,F}$ is a DVR, the multiplicities make sense.) Thus $ X_E$ resembles nothing so much as the projective line over a field! In fact, Fargues and Fontaine show that (again under the assumption that $ E$ is algebraically closed) $ X_E$ is simply connected.

Let me close with an amusing observation. Consider the field $ {\mathbb{C}}_p$, which is of course a perfectoid field. Let $ E={\mathbb{C}}_p^\flat$. (I would call it $ B$, but $ B$ was taken!) Of course one of the un-tilts of $ E$ is $ {\mathbb{C}}_p$, but what are the others? Are they all isomorphic to $ {\mathbb{C}}_p$?

I don’t know. But suppose instead we took $ E$ to be the field of Malcev-Neumann series $ k(\!({x^{\Gamma}})\!)$, where $ k$ is algebraically closed and $ \Gamma$ is a divisible ordered abelian group. Elements of $ E$ are “power series” in $ x$ with coefficients in $ k$ and exponents in $ \Gamma$, where the only restriction is that the support of each power series be a well-ordered subset of $ \Gamma$. You get a valuation on $ E$ by looking at the least exponent of $ x$ that appears in such a series.

$ E$ is a maximally complete field, meaning that any valued extension field $ E’/E$ either has a larger residue field or else a larger value group. There is also a characteristic 0 construction, which I’d like to call $ F=W(k)(\!({p^\Gamma})\!)$, and then $ F^\flat=E$. A result of Bjorn Poonen is that any maximally complete field with residue field $ k$ and value group $ \Gamma$ has to be isomorphic $ F$ or $ E$, depending on its characteristic. (His proof uses the axiom of choice in an essential way.)

(Sometimes I feel that the true $ p$-adic analogue of the complex numbers isn’t $ {\mathbb{C}}_p$ but rather $ F$. The field $ {\mathbb{C}}$ is spherically complete, meaning that any nested sequence of balls has nonempty intersection. $ {\mathbb{C}}_p$ doesn’t have this property, but $ F$ does. Furthermore, elements of $ {\mathbb{C}}$ have decimal expansions. Elements of $ F$ also do, in the sense that every element is a power series in $ p$. How do you write down a generic element of $ {\mathbb{C}}_p$? You basically can’t.)

Let $ (K,\iota)$ be any un-tilt of $ E$. Then $ K$ is maximally complete. (Exercise. Hint: Tilting preserves both residue field and value group.) By Poonen’s result, there is an isomorphism $ f\colon K\to F$. This induces an isomorphism $ f^{\flat}\colon K^{\flat}\to F^{\flat}=E$. Composing $ f^{\flat}$ with $ \iota\colon E\to K^\flat$ gives an automorphism of $ E$ (as a topological field).

This argument shows that $ {\mathrm{Aut} \ } E$ acts transitively on the set of un-tilts $ \left|{Y_E}\right|$. The un-tilt $ F$ of $ E$ gives a point $ \infty\in \left|{Y_E}\right|$, and it is almost tautological to see that the stabilizer of $ \infty$ is the image of $ {\mathrm{Aut} \ } F$ under the natural map $ {\mathrm{Aut} \ } F\to {\mathrm{Aut} \ } E$. Thus there is a bijection

$ \left|{Y_E}\right| \cong ({\mathrm{Aut} \ } E)/({\mathrm{Aut} \ } F). $

Similarly, you get a description of $ \left|{X_E}\right|$ as $ ({\mathrm{Aut} \ } E)/({\mathrm{Aut} \ } F)\phi^{{\mathbb{Z}}}$. I find this rather amazing, since $ {\mathrm{Aut} \ } E$ and $ {\mathrm{Aut} \ } F$ are unimaginably huge groups with no obvious geometric structure, while $ X_E$ is a proper curve. I’m not sure if this observation is useful to the study of $ X_E$, but given the rising role of maximally complete fields in $ p$-adic Hodge theory, it’s worth a look.

Posted in Mathematics | Tagged Fargues, Fontaine, Fundamental Curve, Guest Post, Perfectoid Space, Tilting | 12 Comments

Gerookte paling op de Albert Cuypmarkt

Posted on July 22, 2013 by Persiflage

My mother grew up in, as she would affectionately say, the rat infested slums of Amsterdam (complete with tales of giant rats crawling inside the toilet bowl and sleeping two to a bed). I finally had the chance to visit this half of my ancestral homeland for the first time two weeks ago. My trip was interlaced with the obvious activities (a trip to the Rijksmuseum, exploring the canals) and the less typical ones, including a visit to Dapperbuurt to see where my mother grew up (in a milk bar on Von Zesenstraat).

One connection that people often have with their ancestral culture is via food. That works a little better with the other half of my family (from Bergamo), but still, there were a few traditional Dutch foods that I was looking forward to. Few would say that kroketten scale the heights of culinary achievement, but I was pleased to get to eat some again for the first time in 30 years or so. (As someone who appreciates an Australian meat pie, I am no stranger to food products made with unidentifiable meat sources.) But by far my main culinary desire was in eating smoked eel, which has always been on my list of all-time top ten foods. It’s something that one could buy at the Victoria Market in Melbourne, but the Polish-Ukranian-Russian versions available in the US are over-smoked and over-salted, and so while decent are not quite up to snuff. I found some fillets here, but after a tip from Hendrik Lenstra, I went to one of the markets and found some whole ones at a stall. I bought two and ate them on the spot. That’s not quite accurate, I actually moved a few hundred feet or so away from the market before sitting down and consuming the spoils of my search:

$Delicious Eel$

I spent the second week was at Oberwolfach. I wondered whether it was my third or forth time there — it turned out to be my sixth. There were the usual bridge games with Henri Cohen, Don Zagier, and Mark Watkins, as well as a chess game on the giant board against a mystery opponent (who turned out to be Noam). One analysis of the game suggests that I lost, but an argument can be made that Noam left before the final lecture so that it is a win on time for me. Mark Watkins pointed me towards the following amusing pair of positions:

$win a piece in 517 moves!$

It turns out that this collection of pieces is (typically) a win for white in at most 517 moves. (Here “win” does not mean mate, but rather until White captures one of black’s pieces at which point things become easier.) The two positions above are the board after Black has made his 250th move and 450th move respectively. You are supposed to guess which is which. In other words: you may not be able to compute the best-play sequence of 200 moves, but can you at least tell in which board White is closer to winning? The answer can be found in #370 here.

Next time I’ll talk about the mathematical highlights of my trip, and I promise I’ll get back to Scholze soon.

Posted in Chess, Food, Travel | Tagged Amsterdam, Eel, endgames, Giant Rats, Paling | Leave a comment

Effective Motives

Posted on July 3, 2013 by Persiflage

This is a brief follow up concerning a question asked by Felipe. Suppose we assume the standard conjectures. Let $M$ be a pure motive, and consider the following problems:

Problem A: (“effectivity”) Suppose that $M$ has non-negative Hodge-Tate weights. Then is $M$ effective?
Problem B: (“ordinary primes”) Does the Hodge polygon = Newton polygon for infinitely primes $p$?
Problem C: (“Katz”) Suppose the characteristic polynomials of Frobenius have coefficients in $\mathbf{Z}$. Then is $M$ effective?

An affirmative answer to Problem C implies an affirmative answer to Problem A. Conversely, a positive answer to Problems A & B implies a positive one for Problem C.

The relevance of Problem A was for deducing that a weight zero regular algebraic cuspidal automorphic form for $\mathrm{GL}(2)/F$ could be associated to an abelian variety of $\mathrm{GL}_2$-type over $F$. I claimed that this was probably “Standard Conjectures hard.” It seems that this is partly right and partly wrong.

(Completely unrelated remark: wordpress seems to be a vastly inferior typesetter than LaTeX, since it happily takes LaTeX expressions followed by full stops without a space and separates them by line breaks, and doesn’t even seem to align math formulas within a sentence correctly. Is there a way to integrate the LaTeX more seamlessly into wordpress?)

As mentioned previously, if $M$ has weight zero, then Problem A already follows from Kisin-Wortmann (always assuming the standard conjectures), because then $M$ will be an Artin motive.

As was pointed out to me, the case of weight one follows from the Hodge conjecture. Namely, the Hodge realization gives a polarized Hodge structure of weight one which gives a polarized complex torus. By Riemann, such a torus is actually an an abelian variety $A$, which (using the standard conjectures) one can descend to $F$. This argument doesn’t obviously extend to the general case, because the image of the period map from (say) pure Motives with Hodge-Tate weights $[0,k]$ to polarized Hodge structures will not be surjective for Griffiths transversality reasons. As an aside, it was also pointed out that the Hodge conjecture is not one of the standard conjectures.

When I asked Deligne about Problem A, he politely told me

There’s no evidence for Problem A beyond the fact that it would be nice,
The Hodge conjecture is false, and
Grothendieck already mentioned that his (Grothendieck’s) modification of the generalized Hodge conjecture implies that the answer the Problem A is positive.

Here the generalized Hodge conjecture says (roughly) that a sub-Hodge structure of $H^k$ with weights in the range $[k-q,q]$ to $[q,k-q]$ arises via the Gysin map from an algebraic cohomology class on an $\ge q$-codimensional subvariety. In particular, if $M$ has non-negative Hodge-Tate weights and is of weight $w$, and $M(n)$ is effective inside some smooth proper variety $X$, then $M$ gives rise to a sub-Hodge structure of $H^{w+2n}(X)$ with weights from $[n,n+w]$ to $[n+w,n]$, and hence come from some algebraic subvariety $Y$ of codimension at least $n$. However, the Gysin map on etale cohomology involves a Tate twist by $\mathbf{Q}_p(n)$, and so (using the standard conjectures) one recovers $M$ effectively in $Y$. Grothendieck also points out that, in the case when $M$ has weight one, the generalized Hodge conjecture follows from the usual Hodge conjecture after replacing $X$ by $X \times C$ for proper smooth curves $C$, essentially by the same argument of the previous paragraph. (I guess one also has to use the easy fact that any abelian variety is a quotient of a Jacobian.)

Talking of Deligne and Grothendieck, the Farbster sent me the following link to an interview of Deligne by MacPherson:

https://www.simonsfoundation.org/science_lives_video/pierre-deligne/

which contains the following slightly terrifying exchange about Grothendieck:

MacPherson: I’ve heard people say that he [Grothendieck] was always very kind to students when they didn’t understand, but if someone was older and had pretentions he could be less …

Deligne: That’s quite possible, and I think he was completely willing to explain something once, I don’t think he would have be willing to explain it three times, even to students.

(In my original memory of this passage, “three times” was replaced by “twice.”)

Posted in Mathematics | Tagged Deligne, Farbster, Grothendieck, Motives, Standard Conjectures | Leave a comment

Scholze on Torsion, Part IV

Posted on June 29, 2013 by Persiflage

This is a continuation of Part I, Part II, and Part III.

I was planning to start talking about Chapter IV, instead, this will be a very soft introduction to a few lines on page 72.

At this point, we have reduced the problem of constructing Galois representations for torsion classes on a wide class of locally symmetric spaces to the equivalent problem for Shimura varieties. Naturally enough, the Shimura varieties which arise in this context will not be projective. However, the problem of attaching Galois representations to Hecke actions on $\widetilde{H}^*_c(X)$ is still a very interesting one in the compact case. The difficulties that arise in the non-compact case are somewhat othogonal to the issue of constructing Galois representations, so I don’t think much is lost (at this point) in considering the compact case. (MH tells me that one of the main ingredients for dealing with issues concerning the boundary may well be the Hebbarkeitssatz, II.3.) A good case to keep in mind are the simple Shimura varieties of Kottwitz-Harris-Taylor type, and even the simple case of ball quotients coming from $U(2,1)$ will be of interest. Honestly, even the case of modular curves will be of interest. Modular curves are not compact, of course, but this is the one non-projective case in which the minimal and toroidal compactifications coincide and are smooth, so the boundary causes (relatively) little difficulty. A related problem is to understand the action of Hecke operators on torsion in coherent cohomology. In some sense, Scholze reduces the problem to this case, so we shall begin by considering this problem. Note that already in this case the problem is no longer trivial even for classical modular curves, where one may have torsion in $H^1(X,\omega)$.

Coherent Cohomology: Let $\mathcal{E}$ be an automorphic vector bundle on $X$. Suppose that $X$ is smooth over $\mathbf{Z}_p$, so that it makes sense to impose some nice integral structure on $\mathcal{E}$, and hence to consider the coherent cohomology groups:

$H^*(X,\mathcal{E}/p)$

If $X$ is non-compact, then denote (also by $\mathcal{E}$) the sub-canonical extension to a smooth toroidal compactification. This cohomology group has a natural Hecke action.
How does one construct Galois representations associated to the Hecke action this object? Let’s consider the first non-trivial case, where $X$ is a modular curve and $\mathcal{E} = \omega$. There’s no problem understanding $H^0$, because (via the Hasse invariant) this will be related to classical spaces of modular forms, so the problem is to understand $H^1$. The first step is to understand what $H^1$ is as a vector space. To compute the cohomology of a projective curve, we can take an covering by (two) affines and compute Cech cohomology. To do this, we first need to find two affines. In anticipation of having something sufficiently natural in order to understand the action of Hecke, we let $S$ denote the supersingular locus and $U = X \setminus S$. For now let’s let the other affine be $V$. Then the Cech complex is the following:

$H^0(U,\omega) \oplus H^0(V,\omega) \rightarrow H^0(U \cap V,\omega)$

Here $U$ is the ordinary locus. The space $H^0(U,\omega)$ is the space of ordinary modular forms, and we may relate the Hecke action on this (infinite dimensional) space to the Hecke action on classical modular forms by noting that:

For any section $c \in H^0(U,\omega)$, there exists a power $s^n$ of the Hasse invariant $s$ such that $s^n \cdot c$ extends to $H^0(X,\omega^{m})$ for some integer $m$.
The ordinary locus $U$ is preserved by Hecke operators, and moreover multiplication by the Hasse invariant $s$ is Hecke equivariant.

The problem is that it’s hard to find a second open affine $V$ which is preserved by Hecke, let alone admits an analogue of the Hasse invariant. In this case, we can instead do the following. Take $V$ to be an infinitesimal neighbourhood of $S$, (that is, the completion of $X$ along $S$). Then $V$ is stable by Hecke. Imagine for convenience that there is only one supersingular point. The cohomology $H^0(V,\omega)$ of $V$ has a filtration by the order of vanishing at (each) supersingular point, the first piece consisting of simply functions $H^0(S,\omega)$ on the supersingular point. There exists a section $B^{p-1}$ (see Prop 7.2 of Edixhoven on Serre) which is Hecke equivariant. This approach is used Emerton/Reduzzi/Xiao to construct Galois representations for torsion classes in the coherent cohomology of Hilbert modular varieties (Note that one would also want these representations to satisfy certain local properties at the prime p, which is more subtle in general, but has been done at least for modular curves at least in the residually irreducible case by Calegari and Geraghty.) If one thinks about applying this method in the general case, there are two obvious issues. The first, which is perhaps not impossible to overcome, is that one needs to construct a suitable stratification of the Shimura variety by pieces which one understands and for which one can construct suitable Hasse-invariant type sections which allow one to pass to very ample sheaves whose cohomology vanishes, and hence reduce the problem to degree zero. The second is that, at least in the context of Scholze, one is working at a level which is very ramified at $p$. Certainly all of the discussion above was predicated on $X$ having good integral models at the prime $p$. It’s easier to find good integral models when the corresponding Shimura variety is smooth! At level $X(p^n)$, there do exist integral models (obviously no longer smooth). It’s convenient to assume that the open modular curves $X(p^n)$ are projective, because the issues at the cusps are orthogonal to what is happening here. So what do they look like? Well, they are proper and flat, which is nice. The general problem to the construction is that the torsion subgroup $E[p^n]$ of an elliptic curve $E$ is no longer etale (and so certainly not locally isomorphic in the etale topology to $(\mathbf{Z}/p^n \mathbf{Z})^2$), but it is at least finite flat of rank $p^2$. So all one needs to to is to impose enough extra structure on the finite flat group scheme in order to recover the correct object on the generic fibre and yet have enough points in the special fibre. Katz-Mazur do this by considering a so-called “Drinfeld basis”

$\phi: (\mathbf{Z}/p \mathbf{Z})^2 \rightarrow E[p^n]$

where there is a corresponding equality of Cartier divisors (see 3.1.2 of KM). In particular, given a point $x_n$ one gets a level structure $P_n, Q_n \in E[p^n]$ given by the image of the two generators.

So how does one understand the tower of varieties $X(1) \leftarrow X(p) \leftarrow X(p^2) \ldots$, either integrally or even just on the generic fibre? The ordinary locus up the tower is easy to understand. Let’s first consider the rigid analytic varieties corresponding to the generic fibre. There are sections $X^{\mathrm{ord}}(1) \rightarrow X^{\mathrm{ord}}(p^n)_{\infty}$ from the ordinary locus to the component of the ordinary locus containing infinity, because, for ordinary elliptic curves, we still have etale locally a canonical isomorphism $E[p^n] = \mathbf{Z}/p^n \oplus \mu_{p^n}$, giving an appropriate trivialization. Moreover, the the action of $\mathrm{GL}_2(\mathbf{Z}_p)$ is transitive on the cusps, and so one sees all of the ordinary locus in this way. Thinking more integrally, we can see more directly from Serre-Tate theory that (for all points) at level one the completed local rings will be smooth. However, because $\mathbf{Z}/p^n \oplus \mu_{p^n}$ does not admit any deformations, the covering maps will be smooth at ordinary points and so the complete local rings at any ordinary point will remain smooth. It follows that the interesting geometry will be taking place over the supersingular discs. One can try to understand what is happening by looking at the corresponding completed local rings at supersingular points. Suppose one takes a compatible sequence of supersingular points (in the special fibre) in such a tower. The base point corresponds to a supersingular elliptic curve $E_0$ over $\mathbf{F}_p$ which has a corresponding formal p-divisible group $G_0$, now of height two. What Weinstein teaches us is that whilst the completed local rings $A_n$ of $x_n$ on $X(p^n)$ will be hard to understand, there is still hope to understand the completion

$A = \displaystyle{\widehat{(\lim_{\rightarrow} A_n)}}$

over the ring $\mathcal{O}_K$, which is the completion of $W(\zeta_{p^{\infty}})$. By universality, the Drinfeld level structure gives rise to two parameters $X_n, Y_n$ in $A_n$ which lie inside the maximal ideal. The Weil pairing (we’ve added a consistent sequence of roots of $p$-power roots of unity) gives a relation of the form $\Delta_n(X_n,Y_n) = \zeta_{p^n}$. Jared shows that these are essentially all the relations in the limit ring $A$, which thus has a very nice description. We will come back to this example, because I suspect that understanding this result will be important.

The Lubin-Tate tower There’s also a local analogue of this picture, namely the Lubin-Tate tower. Recall that the Lubin-Tate space $M_0$ is the universal deformation ring of a commutative height $h$ formal group $G_0$ over $k = \mathbf{F}_p$, where $h = 2$. It turns out that $M_0$ is smooth of relative dimension $h-1$ over the Witt vectors $W(k)$. The smoothness is the “same” as the smoothness of the modular curve of level one at a supersingular point. It makes sense to consider level structures in the Lubin-Tate context also, where now the $n$th layer $M_n$ of the Lubin-Tate tower consists of triples $(G,\iota,\alpha)$ with Drinfeld level structure, as in the Katz-Mazur model. Quite explicitly, the $K$-points are given as follows:

$G$ is a formal group over $\mathcal{O}_{K}$,
$G$ is a deformation of the height $h$ formal group $G_0$ over $k$, and $\iota: G_0 \rightarrow G \times k$ is an isomorphism,
$\alpha_n (\mathbf{Z}/p^n \mathbf{Z})^h \rightarrow G[p^n]$ is an isomorphism.

If we go up the entire tower, there is a natural action of $\mathrm{GL}_h(\mathbf{Z}_p)$ in the limit. If $D$ is the corresponding division algebra, then there is an action of $\mathcal{O}^{\times}_D$ on (each) piece of the tower, given by replacing $G$ by a prime-to-$p$ isogeny. In order to have richer actions of $\mathrm{GL}_h(\mathbf{Q}_p)$ and $D^{\times}$ on this tower (not only on the cohomology) it makes sense to modify it slightly (while enlarging the component group in a way that doesn’t change the intrinsic geometry) by considering a trivialization of the rational Tate module $\alpha: (\mathbf{Q}_p)^h \rightarrow V(G)$. Here we now consider deformations up to isogeny, although we remember a quasi-isogeny on a nilpotent divided power thickening of $k$ as well so as not to lose the action of $\mathcal{O}^{\times}_D$. The combined action of these groups on the compactly supported cohomology of the tower realizes the local Langlands correspondence. The proof (for $h = 2$) is to realize this tower geometrically (or at least the cohomology) as the “supersingular part” of the tower of modular curves, and then use global facts concerning automorphic forms. In fact, this is how Harris-Taylor prove local Langlands in general. The corresponding “space” is not literally a rigid space (but more on perfectoid spaces later), but one can ask for a description of the $\mathbf{C}_p$-points of $M$. To this end, one may construct so called period maps. I plan to come back to this in some detail, but for now let me simply say that these maps (constructed in this context in differing contexts and level of generality by Fargues, Weinstein, and Scholze) have their roots in Tate’s $p$-divisible groups paper, where by taking $\mathcal{O}_{\mathbf{C}_p}$-points one may split the $p$-divisible group into a $p$-adic Hodge filtration, and the corresponding period map records the slope of the corresponding line as an element of $\mathbf{P}^1$ (more generally, one obtains a point in a Grassmannian). Let me mention at this point that I have studiously avoided thinking about this whole chapter in the world of Shimura varieties for many years, and it always had the reputation to me as something done by Very Smart People like Mantovan and Fargues, and I have been rewarded in my laziness simply by waiting for the moment where the correct way to view these objects has started to emerge, and there’s someone around like Jared Weinstein who (apart from bringing new ideas) writes and lectures so beautifully well. I certainly recommend reading his papers and lecture notes to understand what is going on (instead of having to sort through the partially digested version I have produced for you here.) Scholze also writes well, thank god.

Page 72: Very roughly, one does the following:

Understand the tower (either the Lubin-Tate tower or the corresponding tower of modular curves) as an actual geometric object $\mathcal{X}$ (perfectoid space).
Construct a period map $\pi: \mathcal{X} \rightarrow \mathbf{P}^1$ (or $\mathscr{Fl}$) using $p$-adic Hodge theory.
Use the first two steps to construct a formal model $\mathfrak{X}$, which will have sections arising via pull-back from some ample line bundle on $\mathbf{P}^1$.
Note that the construction of these sections only depends on the $p$-tower, and so are Hecke equivariant with respect to all the other Hecke operators and can thus serve as a replacement for the Hasse invariant, and multiplication by these sections allows one to pass back to characteristic zero forms in $H^0$, which, by virtue of the control one has over the geometric context, one may identify with classical modular forms.

As Matt explained to me, one can understand the image of the ordinary locus under $\pi$ to be $\mathbf{P}^1(\mathbf{Q}_p)$, which should correspond to the fact that ordinary Galois representations have splittings already before having to pass to $\mathbf{C}_p$. This also fits into the Lubin-Tate story and the period map to the Drinfeld upper half plane (which has $\mathbf{P}^1(\mathbf{Q}_p)$ excised), as occurs in the paper of Fargues linked to above. We also see here that the ordinary locus under the period map factors through the component group $\pi_0$, with the natural action of $\mathrm{GL}_2(\mathbf{Q}_p)$ permuting the cusps. In particular, all the ordinary points are mapping in the special fibre to $\mathbf{P}^1(\mathbf{F}_p)$, which doesn’t look at all like the usual story at all. This is related to footnote #4 on page 72.

Question for the the audience: is it obvious how one can extract the classical coherent cohomology groups $H^*(X,\mathcal{E})$ at level one from $H^*(\mathcal{X}^*,\mathcal{E})$?

Posted in Mathematics | Tagged David Geraghty, Davide Reduzzi, Elena Mantovan, Galois Representations, Jared Weinstein, Langlands, Laurent Fargues, Liang Xiao, Matthew Emerton, Peter Scholze, torsion | Leave a comment

Scholze on Torsion, Part III

Posted on June 22, 2013 by Persiflage

This is a continuation of Part I and Part II.

Before I continue along to section V.3, I want to discuss an approach to the problem of constructing Galois representations from the pre-Scholze days. Let’s continue with the same notation from last time, where $X_M$ is the symmetric space whose cohomology is of interest, and $X = X_G$ is the Shimura variety with Borel–Serre compactification $X^{BS}$ whose boundary contains (simplified assumption: is) a generalized torus bundle $X_P$ over $X_M$. If we localize at a “non-Eisenstein” ideal, then the completed cohomology groups $\widetilde{H}^n(X_M)$ should vanish outside a single degree $q_0$. For this discussion, let us define non-Eisenstein classes to be those which do not occur in degrees $< q_0$ in $H^*(X_M)$. By Hochschild-Serre, any cohomology class in lowest degree (after localization) always survives in the completed limit, so even if one doesn't assume the expected vanishing in higher degrees, the module $\widetilde{H}^{q_0}(X_M)$ will contain all the information about the classes in $H^{q_0}$ at classical level after localization. Hence, to obtain the desired Galois representations for these classes, one wants to prove:

The vanishing of $\widetilde{H}^{q_0-1}(X^{BS})$ after localization.

$\ $

There are Galois representations (of the correct form) associated to classes in $\widetilde{H}^{q_0}_c(X^{BS})$.

$\ $

The hope was that one could try to prove this via the following idealized argument. There is a spectral sequence:

$\mathrm{Ext}^i(\widetilde{H}^{BM}_j,\Lambda) \Rightarrow \widetilde{H}_{d-i-j},$

where $d = 2 \cdot \mathrm{dim}(X)=2n$ is the real dimension of the Shimura variety $X$. There is an identical sequence with the roles of completed homology and completed Borel-Moore homology reversed. Note that the completed homology groups are (Pontryagin dual) to the cohomology groups, which relates compactly supported cohomology to homology and cohomology to Borel-Moore homology. The non-commutative Ext groups in the spectral sequence vanish for any value of $i$ that is less than the co-dimension of the corresponding module. Recall from last time that $\widetilde{H}^*_j$ is torsion except for the middle degree $j = n$. Now suppose that one can show that the completed homology groups $\widetilde{H}^{*}_j$ have sufficiently large co-dimension outside the middle degree. Then from these bounds (and from trivial bounds on the cohomology of the boundary) the spectral sequence should degenerate, and one should have isomorphisms of the following form (after localization):

$\widetilde{H}_{n-i} = \mathrm{Ext}^i(\widetilde{H}^{BM}_n,\Lambda), \ i \le n, \quad \widetilde{H}_{n+i} = 0, \ i > 0, \quad \widetilde{H}^{BM}_n = \mathrm{Hom}(\widetilde{H}_n,\Lambda).$

(To recall, even though we are localizing at an ideal whose avatar on $H^*(X_M)$ is maximally non-Eisenstein, the corresponding ideal on $H^*(X_G)$ will be Eisenstein.) From these equalities, we see that to understand the action of the Hecke operators on completed cohomology, we are reduced to understanding the action on the completed cohomology in middle degree, which we know to be a module of positive rank and hence (even after localization) contain many cusp forms which are known to have interesting Galois representations. At the very least, this would prove the existence of the residual Galois representations associated to such a non-Eisenstein ideal $\mathfrak{m}$. The approach I am outlining here is the one in the (currently non-existent) paper that Matt and I had planned to write. Let’s suppose that one attempts to apply this approach in the Bianchi case. There’s no issue in defining Eisenstein classes here, since the classes that occur in $H^0(X_M)$ are easy to understand, and $q_0 = 1$. So the first step in the above program is to show that $\widetilde{H}^1(X^{BS})$ vanishes, at least if we pass to finite tame level. As we noted last time, this follows from the congruence subgroup property which is known because $U(2,2)$ has real rank two and the corresponding lattice in this group is (obviously) not co-compact. Here the Shimura variety has complex dimension four. So one only has to show that $\widetilde{H}_j$ is small for $j = 2$ and $j = 3$. In particular, one wants, explicitly, that:

$\mathrm{codim}(\widetilde{H}_2) > 4, \ \mathrm{codim}(\widetilde{H}_3) > 3$

The dimension of $\Lambda = \mathbf{Z}_p[[G]]$ is, for reference, $1 + \dim SL_4(\mathbf{Z}_p) = 16$. As noted previously, we know that these cohomology groups are torsion and so have co-dimension at least one. The proof of this result ultimately relied on facts concerning the growth of spaces of automorphic forms. However, it is impossible to determine anything further about the codimension by naïve automorphic considerations, because already $\Lambda/p$ has co-dimension one but no characteristic zero points. So, to prove this conjecture, one really needs to understand the torsion in the cohomology of Shimura varieties. This was where, basically, we were stuck. Note that even understanding $\widetilde{H}^1$ in this case took a powerful result. Understanding $\widetilde{H}^2$ is already much harder. As the real rank increases, it won’t be the case that such completed cohomology groups completely disappear, since there will exist not only trivial stable classes in characteristic zero, but also exotic torsion classes which will be related to K-theory and regulators (as can be seen here). One implication of our conjectures (as noted above) is that the completed cohomology groups vanish for Shimura varieties above the middle dimension. Scholze proves this! (IV.2.3). However, he doesn’t prove it by showing that the $\widetilde{H}_j$ are small for small $j$, and instead deduces a (weaker form) of such an estimate in reverse. I think it’s an interesting problem to understand $H^2(\Gamma,\mathbf{F}_p)$ for groups where the only characteristic zero classes are invariant under $G$, in both the stable and non-stable range. The first case I mentioned previously, and there is something in this direction (in the second case) in section 4.5 of this book.

Section V.3 OK, continuing on from last time, we now have a determinant of dimension $2n$ with image in $A_0/I = \mathbf{T}/I$ for some ideal $I$ with $I^{m} = 0$ for an integer $m$ which only depends on $\mathrm{dim}(X)$. The goal is now to extract an $n$-dimensional determinant, i.e., to recover $\rho$ from $\rho^{\vee} \oplus \rho^c$. Of course, the idea is not to do this from simply one class, but rather allowing twisting, so that we also know $r_{\psi} = \rho^{\vee} \mathrm{det}(\rho) \psi^{-1} \oplus \rho^c \psi^c$ for some Hecke character $\psi$. We may as well take $\psi$ to be a collection of characters of $\mathbf{Q}$, so that $\psi^c = \psi$.

Let’s first make some simplifying assumptions, namely, that the ideal $I = 0$, that we are in characteristic zero, and that the image of $r$ is through a finite group $G$, and the image of all the twists factors through the group $\Gamma:=G \times \mathbf{Z}$ where $\psi$ is a finite order character of the second factor, and $\psi^2 \ne 1$. We would like to imagine that there are equalities:

$r = W =^{?} U \oplus V, \quad r_{\psi} = W_{\psi} =^{?} U \psi \oplus V \psi^{-1}.$

Because the two factors of $\Gamma$ commute, it follows that $[\psi \otimes W_{\psi}] – [W]$
is a virtual character of $\Gamma$. Evaluating this character on the pairs $G \sim (g,1) \subset \Gamma$ defines a class function on $G$. Normalizing by $\psi^2(1) – 1 \ne 0$, this class function applied to $\mathrm{Frob}_{x}$ is the sum of the Satake parameters at $x$ corresponding to $U$, and we deduce that $[U]$, and hence also $[V]$, are virtual characters (with rational coefficients) of $G$. It now suffices to promote $[U]$ to an actual character. The virtual characters $[U]$ and $[V]$ tautologically promote to virtual characters of $\Gamma$ which decompose under the second factor into trivial representations. It follows that $[U \psi]$ and $[V \psi^{-1}]$ are (rational) sums of irreducible representations which decompose under the second factor as direct sums of the representation $\psi$ or $\psi^{-1}$. Assuming that $\psi \ne \psi^{-1}$, there can be no cancellation in $[U \psi] + [V \psi^{-1}]$, from which it follows that $ [U]$ is already an actual character.

In general one has to modify this argument to work more integrally as well as to be compatible with the ideal $J$. As I told TG, “without having looking at this yet, it must essentially be trivial.” So, if you are like me, you can just ignore the following which took me a non-trivial number of hours to work out:

We take the characters $\psi$ to be characters of $\mathrm{Gal}(\mathbf{Q}(\zeta_{\ell^{\infty}})/\mathbf{Q})$ of $\ell$-power order, where $\ell$ is prime to two and $p$ and anything else inconvenient including the ramified primes. This auxiliary prime may vary.

$\ $

Since we are going to allow $\psi$ to have order some arbitrarily large power of various primes, it is convenient to extend scalars to $A = A_0 \otimes W(\overline{\mathbf{F}}_p)$. Here $A_0$ is the Hecke algebra acting with coefficients modulo some fixed power of $p$. It’s useful to work with both rings, however, because whilst $A$ accepts characters of all orders, $A_0$ is literally a finite ring, which is convenient for finiteness arguments. We would like to show that the twisted determinants corresponding to $r_{\psi}$ have values in $A/I_{\psi}$ for the same $A$. This amounts to showing that, at the level of our original locally symmetric quotient $X_M$, we can twist by a sufficiently nice character $\psi$ and not change the Hecke algebra, except for extending scalars. This is straightforward, and is what is going on at the top of page 88.

$\ $

If we have two determinants with a pair of corresponding ideals $I$ and $I_{\psi}$ with $I^m = I^m_{\psi} = 0$, then clearly $\widetilde{I} = I + I_{\psi}$ satisfies $\widetilde{I}^{2m-1} = 0$. So, at the cost of increasing the nilpotency, for any character $\psi$, we get two determinants with values in $A/\widetilde{I}$. Note that if $I$ and $I_{\psi}$ are both trivial, then so is $\widetilde{I}$.

$\ $

We would also like the ideal $\widetilde{I}$ to be independent of $\psi$. Actually, we don’t need this, it will suffice to note that we can take $\widetilde{I} \cap A_0$ to be independent of $\psi$. Because $A_0$ is finite, there are only finitely many such ideals, and so we can take one that occurs for infinitely many primes $\ell$ and infinitely many of the corresponding characters $\psi$.

$\ $

For any fixed character $\psi$, our determinant (which has twice the required dimension) will be defined on a finite quotient of
$\Gamma:=\mathrm{Gal}(L_{\infty}/F) = \mathrm{Gal}(L/F) \times \mathrm{Gal}(F_{\infty}/F),$

where $L/F$ is finite and $L_{\infty}, F_{\infty}$ are the pro-$ \ell$ cyclotomic covers of $L, F$ respectively. This should hopefully look similar to our simplified problem in characteristic zero. We have two determinants $D$ and $D_{\psi}$ with the property that the characteristic polynomial of Frobenius $\mathrm{Frob}_x$ (which exists for determinants) is:
1. Of the form $ P^{\vee}_x(X) P^c_x(X) \mod I$ for $D$.
2. Of the form $P^{\vee}_x(X/\chi(g)) P^c_x(X \chi(g)) \mod I_{\psi}$ for $D_{\psi}$
$\ $

These polynomials $P^{\vee}_x(X)$ and $P^c_x(X)$ are what they obviously should be, namely, the polynomials with inverse roots given by the appropriate Satake parameters. (Or more accurately, with coefficients given by the appropriate Hecke operators.) Because these are determinants, these products are locally constant on the group $\Gamma$ because they are coming from honest Galois representations of rank $2n$. We would like to decompose these into products of two determinants of rank $n$. In the characteristic zero case, we took a character $\psi$ such that $\psi^2(1) – 1 \ne 0$ and used this as a fulcrum on which to tease out the representation $U$. Here we do something similar. A first step is to show that each of the four polynomials above is locally constant. We choose an element $1 \in \mathrm{Gal}(F_{\infty}/F)$ and a deep enough character $\chi$ so that $\chi^{2m}(1) -1 \ne 0$ for all $m = 1,\ldots,n$. We now find an open neighbourhood of $(G,1)$ where $D$ and $D_{\chi}$ are constant. Let $a(x)$ be the linear term of $P^{\vee}_x(X)$, and let $b(x)$ be the linear term of $P^c_x(X)$. Then we deduce that the following two terms are locally constant:

$a(x) + b(x), \quad a(x) \psi(x) + b(x) \psi^{-1}(x).$

So, because $\psi^2(x) -1 \ne 0$, we deduce that $a(x)$ and $b(x)$ are locally constant, and so $a(x) \mod \widetilde{I} \cap A_0$ is also locally constant. Given this, one proves that the quadratic terms are also locally constant in the same manner, and by induction one has the result for the entire polynomial. Thickening the open neighbourhood up, one proves the same result for the entire group $\Gamma$ minus the piece coming from $\mathbf{Z} \sim \mathrm{Gal}(F_{\infty}/F)$, which gives us Lemma V.3.4. Then by choosing a different auxiliary prime $\ell’$, one patches to get a well defined class function on $G$ in Lemma V.3.5.

$\ $

So now we have a class function $D$ on the Galois group $\mathrm{Gal}(L/F)$ with values in characteristic polynomials (now of the right dimension!) in $A_0/I$ (dropping the tildes), and we want to promote it to a genuine determinant. Of course, over finite rings we can’t use the language of virtual characters. What Scholze does next is use the fact that we have such a decomposition for infinitely many different characters in order to glue enough of them together to obtain a determinant map
$D: A[G \times \mathbf{Z}] \rightarrow A[t]/I, \quad D(1 – X \gamma^k g) = P^{\vee}_g(X/t^k) P^c_g(X t^k),$

where $\gamma$ is a generator of $\mathbf{Z}$ and $I$ has the expected properties of nilpotence. This consists of Lemmas V.3.6 and V.3.7.

$\ $

Now we are at Lemma V.3.8. Bugger it, this is taking a long time, and quite possibly nobody is interested in these specific details. Let me cut some corners and replace determinants by pseudo-representations. We deduce from the above that we are in the following situation: we have a degree $2n$ pseudo-representation:
$T: G \times \mathbf{Z} \rightarrow A[t], \qquad T(g,m) = a(g) t^m + b(g) t^{-m}.$

We want to deduce that $a(g)$ and $b(g)$ are both pseudo-representations of degree $n$. We are allowed to use the fact (which is obvious) that $a(g)$ and $b(g)$ are not pseudo-representations of degree strictly less than $n$. (Actually, is it obvious? It’s certainly obvious for $n = 2$ that $a(g)$ and $b(g)$ are not a character. So let’s assume $n =2$. Ah, I see by passing to the trivial element we can compute that $a(1) = b(1) = n$, so it is obvious.) Now, if we abstract slightly and drop any knowledge about $a(g)$ and $b(g)$ other than they are class functions, the best we can hope to prove is that $a(g)$ and $b(g)$ are both pseudo-representations of degrees $A$ and $B$ respectively, where $A+B=2n$. This is what we do. Since $T$ is a pseudo-representation of degree $2n$, we have the following identity:

$\displaystyle{\sum_{S_{2n+1}} (-1)^d T_{\sigma}(g_i,m_i) = 0.}$

In fact, this identity on class functions characterizes pseudo-representations of degree at most $2n$, the only other information coming from evaluating on the identity. Suppose we take the $m_i$ to be sufficiently generic integers so that all the sums $\sum \pm m_i$ are distinct. Now let us partition the $m_i$ into two sets $M_A, M_B$ of cardinality $A+1$ and $B$ respectively, where $A+B=2n$. Consider the coefficient of $t^C$ in the sum above, where we take

$C:= \sum_{M_A} m_i – \sum_{M_B} m_i$

The corresponding coefficient must vanish. Moreover, because of the way that the $m_i$ were chosen, we know exactly what terms can arise with this coefficient: explicitly, the terms in $M_A$ must come from $a(g)$, and the terms in $M_B$ must come from $b(g)$. Hence we recognize the coefficient to be (up to sign)

$\displaystyle{\left(\sum_{S_{A+1} \curvearrowright M_{A}} (-1)^d a_{\sigma}(g_i) \right) \left( \sum_{S_B \curvearrowright M_{B}} (-1)^d b_{\sigma}(g_i) \right)}.$

We deduce that, for any decomposition $A+B=n$, either $a(g)$ is a pseudo-representation of degree at most $A$, or $b(g)$ is a pseudo-representation of degree at most $B-1$. Taking $B$ to be the smallest integer for which $b(g)$ is a pseudo-representation, we deduce the result (such an integer exists because $b(g)$ is at least a degree $\le 2n$ pseudo-deformation). We are, mercifully, done. Looking at Scholze, I think this lemma (and even roughly the argument) is quite similar to the proof of Lemmas V.3.8-V.3.15 but this is much easier, at the cost of assuming that $p > n$.

It looks as though one can probably skip step 6 simply by choosing the value of $t \sim \psi(1)$ to generate a sufficiently generic extension of $A_0$ inside $A$, although I guess that’s how one does step 6 anyway.

Section V.4 is just a matter of putting things together. Next time: onto Chapter IV!

Posted in Mathematics | Tagged Galois Representations, Langlands, Peter Scholze, torsion, Vaporware | Leave a comment

Scholze on Torsion, Part II

Posted on June 19, 2013 by Persiflage

This is a sequel to Part I.

Section V.1:
Today we will talk about Chapter V. We will start with Theorem V.1.4. This is basically a summary of the construction of Galois representations in the RACSDC case, which follows, for example, from work of Shin. We know a little bit more than this theorem states (namely, local-global compatibility).

Corollary V.1.7 is just the statement that the cohomology groups $H^0(X,\mathcal{E}_{\mathrm{sub}})$ for the sub-canonical extensions of automorphic sheaves $\mathcal{E}$ are computed by forms $\pi$ whose transfer to $\mathrm{Res}_{F/\mathbf{Q}}\mathrm{GL}(n)$ are RACSDC. The sub-canonical extension corresponds to imposing a vanishing condition at the cusps. For example, the sub-canonical extension of $\omega^k$ on the modular curve is $\omega^k(-\infty)$. There is a nice action of the Hecke algebra $\mathbf{T}$ on this space, which is compatible with the associated Galois representations in all the expected ways (Satake parameters to Frobenius eigenvalues) at the unramified primes. So far, this is all classical (as of 2011).

Determinants: We will be using congruences to obtain Galois representations, but the information that gets glued is really the Hecke eigenvalues. So one wants a convenient way to pass from one to the other. The classical approach with modular forms is to remember the “standard” Hecke operators $T_x$ which correspond to the traces of Frobenius. Knowing the trace is enough to determine a two dimensional representation away from characteristic $2$, if one has residual irreducibility. This is the theory of pseudo-representations. Naturally enough, for larger dimensional Galois representations, it helps to remember more than the trace, namely, the entire characteristic polynomial. The corresponding theory was worked out by Chenevier. Namely, given an $n$-dimensional representation $\rho$ of the group $G$ over a commutative ring $A$, there is a map:

$D: A[G] \rightarrow M_n(A) \rightarrow A$

given by formally extending $\rho$ in the obvious way and then composing with the determinant. For example, if $n = 2$, then

$T(g):=D([g] + [1]) – D([g]) – D([1]) = \mathrm{Tr}(\rho(g)).$

Now the map $D$ has to satisfy a bunch of formal properties due to the constraints of coming from an $n$-dimensional representation. Writing all these down gives the correct notion of Chenevier’s generalized “determinant.” (Original paper here.) For those who like pseudo-representations, note that when $n=2$, one can define $D$ using the formula:

$D(g) = \displaystyle{\frac{T(g)^2 – T(g^2)}{2}},$

where $T$ is the trace. So for $n = 2$ in characteristic greater than two, the notions are equivalent. And indeed Chenevier’s notion of determinants is the same as a pseudo-representation whenever $n!$ is invertible, but is better behaved in small characteristics. Determinants satisfy the nice properties that pseudo-representations do, and that Galois representations sometimes don’t (but do in the residually absolutely irreducible case), namely:

You can glue determinants: $D \rightarrow A/I$ and $D \rightarrow A/J$ which agree on $A/(I + J)$ to get a determinant $D \rightarrow A/(I \cap J).$
Given a formal variable $X$, there is a natural determinant map $D: A[X][G] \rightarrow A[X]$ such that $D(1 – X g)$ is the characteristic polynomial of $g$ if the determinant comes from an actual representation.

Here I follow Scholze in using $\mathrm{Det}(I – X \cdot M)$ rather than $\mathrm{Det}(M – I \cdot X)$ as the definition of a characteristic polynomial — this is just a bookkeeping issue (the dreaded arithmetic versus geometric Frobenius). Returning to automorphic forms from coherent cohomology, since $H^0$ is torsion free, the module $\mathbf{T}$ is flat over $\mathbf{Z}$. Since the characteristic zero forms give rise to Galois representations coming from RACDSC forms, we naturally obtain a determinant map:

$D: \mathbf{Z}_p[G_{F}] \rightarrow \mathbf{T}$

such that $D(1 – X \cdot \mathrm{Frob}_x)$ is exactly as one would expect. (This is Corollary V.1.11). Note that the ring $\mathbf{T}_c$, which arises at this point, is just the inverse limit of the corresponding classical $\mathbf{T}$ over all $p$-power levels; this is defined in Chapter IV which we shall talk about later.

Segue on Completed Cohomology: I have to recall here a few basics about completed cohomology (one reference is here.) I already know about completed cohomology (and so do many of my loyal readers) so I don’t really feel obliged to say too much about it, but since most of you have been sent here from Quomodocumque, I will cough up a few pointers. The basic definition (for any congruence arithmetic manifold corresponding to a group $\mathbb{G}$) is as follows:

$\widetilde{H}^i(X,\mathbf{Z}/p^n \mathbf{Z}) := \lim_{K \rightarrow} H^i(X(K),\mathbf{Z}/p^n \mathbf{Z}).$

Here the limit is over shrinking compact open subgroups $K$ of $\mathbb{G}(\mathbf{Z}_p)$. The tame level is fixed and can be included in the notation somewhere. One can also adorn the cohomology groups in the usual way, namely, by considering compactly supported cohomology. So what’s the point of completed cohomology? Apart from having a natural action of $\mathbb{G}(\mathbf{Q}_p)$, which is always the type of group one wants to act on a candidate space for automorphic representations of any kind, a matter of experience and intuition suggested (to Matt and me) that it should be the “correct” space of automorphic forms modulo $p^n$ when $\mathbf{G}(\mathbf{R})$ does not have discrete series (and even when it does). One way to justify this is via the following four properties, the final one conjectural:

The completed cohomology groups $\widetilde{H}^i(X,\mathbf{Z}/p \mathbf{Z})$ are co-finitely generated over $\Lambda = \mathbf{F}_p[[\mathbb{G}(\mathbf{Z}_p)]]$. This latter ring has nice properties, e.g. after shrinking the group $\mathbb{G}(\mathbf{Z}_p)$ slightly to get a powerful torsion free pro-$p$ group, $\Lambda$ is a local Noetherian ring which is Auslander regular (see Lazard and also Venjakob.)

$\ $

The Pontryagin dual groups $\widetilde{H}^i(X,\mathbf{Q}_p/\mathbf{Z}_p)^{\vee}$, which are finitely generated (by part one and Nakayama’s Lemma and the usual long exact sequences) are not torsion $\Lambda = \mathbf{Z}_p[[\mathbb{G}(\mathbf{Z}_p)]]$-modules if and only if one is in middle degree and the corresponding real group admits discrete series (see this paper).

$\ $

The completed homology groups satisfy a Poincaré duality spectral sequence. The completed cohomology groups are compatible with the Hochschild–Serre sequence from which one can recover classical cohomology groups.

$\ $

Given a torus bundle, or more generally a nilmanifold, the completed cohomology disappears outside degree zero.

$\ $

Conjecturally: for any reductive algebraic group there will be a dominating term $\widetilde{H}^i(X)$ in degree $i = q_0$ which will have co-dimension $l_0$ as a $\Lambda$-module, where $2 q_0 + l_0$ is the real dimension of $X_G$, and the degrees $[q_0,q_0+1,\ldots,q_0 + l_0]$ are exactly the degrees in which tempered automorphic representations contribute to cuspidal cohomology. More directly, $l_0$ for a semi-simple group is the rank of $\mathbb{G}(\mathbf{R})$ minus the rank of the maximal compact. For example, $l_0$ is equal to zero if and only if the real group admits discrete series. Hence this bullet point is a conjectural generalization of point (2). As an example, in the case of $\mathrm{GL}_2$ over an imaginary quadratic field, the completed cohomology $\widetilde{H}^1(\mathbf{F}_p)$ should have codimension exactly one.

$\ $

(For the last three points I’ll refer you once again to this survey.)

Section V.2: The key starting point, as mentioned last time, is that one can relate the cohomology of the group we are interested in — $\mathrm{Res}_{F/\mathbf{Q}}(\mathrm{GL}_n)$ — to the cohomology of Shimura varieties by realizing the first group as the Levi $M$ inside a maximal parabolic $P$ inside a group $G$ corresponding to a Shimura variety. The first step is to compare the cohomology of what we are interested in (coming from the Levi $M$) to the cohomology of the boundary piece coming from the parabolic $P$ inside $G$ containing $M$. This is pretty standard: what happens is that the resulting space $X_P$ which actually occurs in the boundary of the Borel–Serre compactification $X^{BS}_G$ of $X_G$ is a torus bundle over $X_M$. Well, not literally always a torus bundle, but rather a nilmanifold $N$ coming from the unipotent part of $P$. The nilmanifold fibres spread the cohomology around by a Künneth type formula like a Frenchman expectorating over-oaked California Chardonnay into a spittoon. (Usually this fibration arises as a quotient from a fibration with a contractible fibre, which means that the cohomology really is just the derived product of the cohomology of the base and the cohomology of $N$, so it’s not really so bad.) One way to avoid this mess is by passing to completed cohomology. On the boundary this has the effect of collapsing all the torus like directions in the nilmanold, and obtaining a map from the completed cohomology of the arithmetic manifold corresponding to the Levi into the completed cohomology of the total space. Compare with equation (1.4) of this survey again.

Hecke Operators from $M$ to $G$. One thing we have to understand is how to compute the Hecke operators at unramified primes on the completed cohomology of the boundary of $X_G$ in terms of the action of the Hecke operators on the original object of interest $X_M$. Let us fix an unramified prime $x$ which is prime to everything. To orient you, we are at the top of page 82 of Scholze. I’m going to be more prosaic in my notational choices and write $T_G$, $T_P$, and $T_M$ for the local Hecke algebras at the prime $x$ (Scholze does all the unramifed primes at once). Yes, I know this is an abuse of notation, because here the groups $G$, the parabolic $P$ and the Levi $M$ are really the local versions at the prime $x$. (You will cope.) There are natural maps:

$T_G \rightarrow T_P \rightarrow T_M.$

Let’s actually consider what these are in the case when $ M$ comes from $\mathrm{GL}(2)$ over an imaginary quadratic field $F$ in which $x$ splits, and $G$ comes from $U(2,2)$ which also splits over $F$. So locally at $x$, the group $G$ is just $\mathrm{GL}(4)$, and $M$ is the levi $\mathrm{GL}(2) \times \mathrm{GL}(2)$, and $P$ is what it obviously has to be. In this case, we have isomorphisms:

$T_G \simeq \mathbf{Z}_p[X^{\pm}_1,\ldots,X^{\pm}_{4}]^{S_4}, \qquad T_P \simeq \mathbf{Z}_p[Y^{\pm}_1,Y^{\pm}_2]^{S_2} \times \mathbf{Z}_p[Z^{\pm}_1,Z^{\pm}_2]^{S_2}.$

Perhaps we are required to adjoin $\sqrt{x}$ to both sides in order to normalize this appropriately. Consider it done. Now the map $T_G \rightarrow T_{M}$ is the one sending $(X_1,X_2,X_3,X_4) \mapsto (x Y_1,x Y_2,x^{-1} Z_1,x^{-1} Z_2)$. The choice here must be coming from the choice of $M$ (for a fixed torus) corresponding to a choice of subgroup $S_2 \times S_2$ of the Weyl group. One can write down analogous formulas for the inert and ramified primes. The corresponding maps of Satake parameters indicates that the if our original eigenclass has a Galois representation $\rho$, then the Hecke eigenvalues of the class which has been pulled back is associated to $\rho^{\vee} \oplus \rho^c$. (Edit: In the previous version I omitted the dual. Note that $\rho^{\vee} \det(\rho) = \rho$ for $n = 2$. End Edit) Now this statement seems to be somewhat in conflict with my previous post, where I claimed that the action of the Hecke algebra on the cohomology of $U(2,2)$ corresponded to the Galois representation $\rho \otimes \rho^c$. This is because of a subtlety which I think I can explain. Suppose you start from a classical modular form $f$ and base change it to a Hilbert modular form $f_E$ over a real quadratic extension. Then the corresponding map of Satake parameters is just the obvious one corresponding to the restriction of the Galois representation. In particular, if $\alpha,\beta$ are the Satake parameters of a local unramified component $\pi_x$ of $f$, and if $x$ splits in the quadratic field and $y$ is a prime above $x$, then $\pi_y$ of $f_E$ will have the same Satake parameters, and $f_E$ will have the same Hecke eigenvalue for $T_y$ that $f$ has for $T_x$. However, the actual Galois representation occurring inside the etale cohomology of the Hilbert modular surface is not the restriction of the Galois representation to $E$, but rather the (four dimensional) tensor induction. This also reflects an important point: we will not be finding the desired Galois representation inside etale cohomology (which, apparently by an argument of Clozel and Harris, is impossible), but rather we will simply be “following the Hecke eigenclasses.” In this context, for example, cuspidal automorphic representations for $U(2,2)$ contain all the information for the associated four-dimensional representations, but the ones occurring in cohomology are (tensor inductions!) of $\wedge^2$. That is why in this post we see the Hecke eigenvalues as looking like the direct sum $\rho^{\vee} \oplus \rho^c$, whereas the action on cohomology via Eichler-Shimura looked like $\wedge^2 (\rho^{\vee} \oplus \rho^c)$, which contains $\rho \otimes \rho^c$ up to twist.

The arguments on the lower half of page 82 are just related to the fact that the boundary of the compactification on $X_G$ can have a number of components, and these components can have their own boundary, and so on. If one takes the case where $X_G$ corresponds to $U(2,2)$ over an imaginary quadratic field, then the only boundary components are (torus bundles over) Bianchi manifolds $X_M$, and the only boundaries that they have are hyperbolic cusps. In particular, in this case, using remark (4) on completed cohomology above, the completed cohomology of the boundary $\widetilde{H}^k(\partial X_G)$ (denoting $X^{BS}_G \setminus X_G$ by $\partial X_G$), is given by

$\widetilde{H}^k(\partial X_G,\mathbf{Z}/p^n \mathbf{Z}) = \mathrm{Ind}^{\mathbb{G}(\mathbf{Z}_p)}_{\mathbb{P}(\mathbf{Z}_p)} \left(\widetilde{H}^k(X_M,\mathbf{Z}/p^n \mathbf{Z})\right).$

So we are interested in the Hecke action on the right hand side, which we have now transferred to the left hand side. (Of course, the local Hecke algebras combine by taking tensor powers to get the Hecke algebra at all unramified primes, which surjects onto the corresponding global Hecke algebras $\mathbf{T}_G$ and $\mathbf{T}_M$.) There is a natural long exact sequence of completed cohomology associated to a manifold with corners as follows:

$\ldots \rightarrow \widetilde{H}^{k-1}(X_G,\mathbf{Z}/p^n \mathbf{Z}) \rightarrow \widetilde{H}^{k-1}(\partial X_G,\mathbf{Z}/p^n \mathbf{Z}) \rightarrow \widetilde{H}^{k}_c(X_G,\mathbf{Z}/p^n \mathbf{Z}) \rightarrow \widetilde{H}^k(X_G,\mathbf{Z}/p^n \mathbf{Z}) \rightarrow \ldots $

So to get a Galois representation (or, to begin with, a determinant) on $\widetilde{H}^{k-1}(\partial X_G)$, we can start by finding determinants for the two surrounding terms.

Special Case: Let’s continue discussion the special case where $X_M$ is a Bianchi manifold, and $X_G$ comes from $U(2,2)$ which splits over the corresponding imaginary quadratic field. The key term of interest will be (for the Bianchi manifold) $\widetilde{H}^1(X_M)$ or, equivalently, $\widetilde{H}^1(\partial X_G)$. In fact, by Hochschild-Serre, the completed cohomology $\widetilde{H}^1(X_M)$ captures all the interesting Hecke actions coming from torsion in Bianchi groups as long as one localizes away from the Eisenstein primes coming from the cusps. The cusps in the Borel–Serre compactification of the Bianchi group are elliptic curves with CM by the underlying imaginary quadratic field. The difference between the classical classes in $H^1$ and $\widetilde{H}^1$ proved themselves to be a real pain in my book with Akshay, because when one wants a numerical correspondence, one can’t ignore Eisenstein terms. Yet blessedly, in this context, we can localize away from them. Hence the key terms are those in the following boundary exact sequence:

$\widetilde{H}^{1}(X_G) \rightarrow \widetilde{H}^1(\partial X_G) \rightarrow \widetilde{H}^{2}_c(X_G)$

Let’s consider the first term. The group $U(2,2)$ has real rank two. In particular, ~~by super rigidity~~, any non co-compact lattice in $U(2,2)$ will have the congruence subgroup property. It follows that $\widetilde{H}^1(X_G)$ is trivial! The point is that if all the finite quotients of a lattice in $U(2,2)$ come from congruence quotients, then pulling back over all such quotients kills everything. Actually, this is not strictly correct, because completed cohomology only pulls back over $p$-power quotients, and there may be cohomology coming from the tame level. However, it is easy to see (by Hochschild–Serre) that any such cohomology will be Eisenstein. In particular, after localizing at a non-Eisenstein (in the appropriate sense) ideal, we get an injection from $\widetilde{H}^1(\partial X_G)$ to $\widetilde{H}^{2}_c(X_G)$, and thus from Theorem IV.3.1, we obtain a determinant to the Hecke algebra of $\widetilde{H}^1(X_M)$ (localized away from Eisenstein ideals) without any need to quotient out by an ideal with fixed zero power as in Corollary V.2.6. I don’t think this trick will really work in any other examples, however, since it’s very hard to say anything in general about $~H^2$. (There is recent work on on stable completed co/homology here, but that will never be enough to give something useful in this context.)

General Case: The general case is now quite similar, except know to understand $\widetilde{H}^k(\partial X_G)$ one needs to understand both boundary terms. There is also going to be some loss of information coming from the corresponding extension class. If one had determinants on $\widetilde{H}^*(X_G)$ and $\widetilde{H}^*_c(X_G)$, then one would immediately get Corollary V.2.6 with an ideal $I$ with $I^2 = 0$. However, Theorem IV.3.1 (which is being invoked here) only applies to $\widetilde{H}^*_c(X_G)$. Now $\widetilde{H}^*(X_G)$ is related to its compact cousin by a Poincaré duality spectral sequence, but this will once again spread out some terms and necessitate replacing $I^2 = 0$ by some power involving the dimension. At any rate, while there is room for improvement in general, there is still the fundamental problem (mentioned in part zero!) of controlling whether this boundary cohomology is going forwards or backwards in the long exact sequence above (or worse, being mixed). I’m going to give some heuristics next time on what one expects should happen (short answer: after localizing at a nice maximal ideal, it should work out as well as the Bianchi case, but that will be hard to prove.)

Note that Scholze actually works with classical cohomology here, and then relates it back to completed cohomology using Hochschild-Serre on p.86. The point in either argument is that all the terms in the spectral sequence (on every page) are, by Theorem IV.3.1, modules for the Hecke ring $\mathbf{T}_c$ which acts on coherent cohomology. Hence the limit terms have filtrations by a fixed bounded number of such objects.

Next time: I’ll say a little more about how one might expect the “simplification” in the Bianchi case above to apply more generally, and I’ll talk about the final section V.3 of chapter V, in which we extract the $n$-dimensional representations from our $2n$-dimensional determinants.

Posted in Mathematics | Tagged Bianchi Groups, Determinants, Gaëtan Chenevier, Galois Representations, Peter Scholze, torsion | 6 Comments

Scholze on Torsion, Part I

Posted on June 16, 2013 by Persiflage

This is a sequel to this post, although as it turns out we still won’t actually get to anything substantial — or indeed anything beyond an introduction — in this post.

Let me begin with some overview. Suppose that $X = \Gamma \backslash G/K$ is a locally symmetric space, where $G$ is a semi-simple group which does not admit discrete series. To be concrete, suppose that $G = \mathrm{SL}_2(\mathbf{C}),$ and that $\Gamma$ is an congruence subgroup of level $N$ in the Bianchi group $\mathrm{SL}(\mathcal{O}_F)$ — recall here that $F$ is an imaginary quadratic field. Since days of yore (Langlands, Clozel, Fontaine-Mazur, etc.), everyone has expected that there should exist a bijection between the following objects:

Regular algebraic cuspidal autormophic representations $\pi$ for $\mathrm{GL}_2(\mathbf{A}_F)$ of level $N$ and weight zero (= the same infinitesimal character as the trivial representation).
Cuspidal (= non-boundary) cohomology classes $H^1(\Gamma,\mathbf{R})$ which are eigenforms for the ring of Hecke operators $\mathbf{T}$.
Weakly compatible families of two-dimensional Galois representations of $F$ which are irreducible of level $N$ and Hodge-Tate weight $[0,1]$.
Irreducible semi-stable $p$-adic Galois representations of weight $[0,1]$ and level (determined in the usual way) dividing $N$.
Abelian Varieties of $\mathrm{GL}(2)$-type over $F$ which don’t have CM by $F$.

The equivalence between $(1)$ and $(2)$ follows from Matsushima/Franke. The (conjectural) relationship between $(1)$ and $(3)$ is the problem of reciprocity in the Langlands programme. It consists of two directions; existence (i.e. constructing Galois representations from automorphic forms) and modularity (i.e. showing nice Galois representations are automorphic). Both of these directions are difficult. The passage from $(5) \Rightarrow (3) \Rightarrow (4)$ is easy, whereas $(4) \Rightarrow (5)$ is basically the Fontaine-Mazur conjecture (not so easy). Actually even that last statement is not quite correct: if one knows that $V$ is pure of weight one with non-negative Hodge-Tate weights and arises up to twist in the cohomology of some smooth proper algebraic variety, then it should actually arise in cohomology without having to twist and hence come from an Abelian variety; proving this, however, is probably hard (as in Standard conjectures hard). Note that $(1) \Rightarrow (5)$ follows, in the case of classical modular forms, from a geometric construction of Shimura, but that idea doesn’t work here, and in fact this arrow is completely open and we shall say no more about it. In the particular case of imaginary quadratic fields, Harris, Soudry, and Taylor showed in 1992 that $(1) \Rightarrow (3)$ under certain favourable conditions. This case is slightly exceptional in this regard, since there exist functorial transfers of such $\pi$ to $\mathrm{GSp}(4)$ which do contribute to the $(\mathfrak{q},K)$-cohomology of Shimura varieties and hence can be directly related to the coherent cohomology of Shimura varieties (although not directly to Betti cohomology, because the corresponding weights are not regular.) As readers of this blog know, only very recently, Harris-Lan-Taylor-Thorne established the same result for $\mathrm{GL}_n$ over a CM field. (Small lie: not all the desired properties of the corresponding Galois representations, — i.e. local-global compatibility — have been established. I think Ila Varma is working this out for her thesis.)

It was observed early on that the cohomology groups of $X$ are not, in general, torsion free. So what then does a torsion class represent? Computations by Grunewald, Helling, and Mennicke in an 1978 paper suggested that torsion classes (specifically, two-torsion with Hecke eigenvalues in $\mathbf{F}_2$) in these groups should be associated to $\mathrm{GL}_2(\mathbf{F}_2) = S_3$ Galois representations of the field $F$. Apparently there are even some unpublished notes from Grunewald in 1972 doing similar things, although I have only ever heard rumors of their existence (to be fair, I heard those rumours from Grunewald, so they’re probably pretty reliable). So very early on there were hints that a further story was going on between torsion classes and Galois representations that wasn’t immediately related to the (conjectural) story coming from automorphic forms. The first general and precise conjecture along these lines were formulated by Ash in his 1992 Duke paper, with further refinements by Ash–Sinnott, and Ash–Doud–Pollack. Unlike the previous speculations of Grunewald, these conjectures were precisely formulated and falsifiable, and in the spirit of Serre’s original conjecture. Moreover, Herzig did actually come along and falsify them, by finding a more natural prediction for the set of possible Serre weights which turned out to be different from the formulation of Ash et. al., and Herzig’s formulation subsequently proved (numerically!) to give the right answer in these cases (Edit: see comments, this is not quite correct). At any rate, for quite some time, we have expected that mod-p torsion classes give rise to Galois representations, and following the conjectures of Serre, Ash, and others, one can be quite precise about exactly the local properties the corresponding Galois representations should have at primes of bad reduction. What is perhaps more recent is the idea that, especially for groups $G$ with no discrete series, that torsion is not merely a techincal nuisance, but rather is the source of “most” of the interesting Galois representations. In particular,

The phenomenon whereby Galois representations coming from the countably many classical automorophic forms are dense in a suitable universal deformation ring (Böckle, Gouvêa-Mazur, Chenevier) will be totally false when $G$ does not have discrete series. On the other hand, the representations coming from torsion should cut out all of the universal deformation ring.
That in order to answer the most pressing questions concerning reciprocity (even in characteristic zero!) one needs to understand torsion classes.

$\qquad$

For one take on this, I might suggest reading Section 1.1, Speculations on
p-adic functorality, of this paper. For another interesting perspective, you should also read this, as well as the accompanying review.

So let us assume then that studying torsion representations and associating them to Galois representations is an Important Goal. How do we construct them? An observation also going back a long way (I believe to Harder??) is the following. Even though $G$ may not admit discrete series, there may exist a group $H$ containing a parabolic $P$ with $G$ as a Levi. If $H$ does admit discrete series, then there will exist a Shimura variety $X_{H}$ whose Borel-Serre compactification will have at least one boundary component which is a torus bundle over $X_{G}$, and as a result one obtains a map (with some mixing of degrees) $H^*(X_{G},\mathbf{Z}) \rightarrow H^*(\overline{X}_{H},\mathbf{Z})$. Now one is theoretically in better shape, because this map should be compatible (in some sense) with Hecke operators, and the latter group has a chance to admit comparisons to étale cohomology groups which do come with Galois representations. There are three immediate problems:

The compactification $\overline{X}_{H}$ will be singular, except in the case of modular curves.
Given a long exact sequence for (co)homology relative to a boundary divisor, it’s not clear whether the cohomology in the boundary ends up in $H^{i-1}$ or $H^{i}$.
Just because a class $[c]$ has an interesting Hecke eigensystem doesn’t mean that that étale cohomology sees an interesting Galois representation.

$ \ $

The third issue is a genuine problem. If one has a Hecke eigenclass $[c] \in H^*(X)$ in the etale cohomology of a Shimura variety, even in characteristic zero, then all one can deduce is that the corresponding Galois representation is annihilated by the corresponding characteristic polynomial of Frobenius. But this is not always enough to get the correct Galois representation! It’s probably worthwhile to consider two examples.

First, a somewhat degenerate example. Let $\mathbb{G} = \mathrm{GL}(1)/\mathbf{Q}$, $G = \mathbf{R}^{\times}$, $\mathbb{H} = \mathrm{SL}(2)/\mathbf{Q}$, and $H = \mathrm{SL}_2(\mathbf{R})$. Now $X_{G}$ is (for some level structure) a finite set of points, and $X_{H}$ is a modular curve with cusps. The boundary map realizes $X_{G}$ as a set of cusps in $X_{H}$. The Hecke operators act on cusps by the degree map, i.e. $T_p[c] = (1+p)[c]$. This coincides with the action of Hecke operators on $H^0(X_H)$. So, in the etale cohomology group $H^0(X_H)$ we have a Hecke eigenclass which we imagine (looking at the Hecke operators) to be associated to the Galois representation $1 \oplus \chi$ where $\chi$ is the cyclotomic character. Yet $H^0(X_H)$ is one dimensional, and so we only see half of the Galois representation, namely, the trivial character. Now as it turns out, the other piece of the Galois representation can be seen in $H^1(X_H)$, which is now mixed because $X_H$ is not projective (so the cuspidal part has motivic weight one, and this other piece of the Eisenstein series has weight two). So even in this trivial case, we see that a Hecke eigenclass in etale cohomology may have a less interesting Galois representation than the Hecke eigenvalues might suggest. From the Eicher-Shimura relation, we do get that the (trivial) Galois representation which does occur is annihilated by the characteristic polynomial of Frobenius $(\sigma – 1)(\sigma – \chi(\sigma))$, indeed it is annihilated by the first factor.

Second, let $\mathbb{G} = \mathrm{GL}(2)/F$, $G = \mathbf{GL}_2(\mathbf{C})$, $\mathbb{H} = U(2,2)/\mathbf{Q}$, and $H = U(2,2)$. Here $\mathbb{H}$ is taken to split over $F$. The cohomology of (a torus bundle over) the Bianchi group maps into the cohomology of $U(2,2)$. The characteristic polynomials of the Hecke operators are, morally, the following. If $\rho$ is the (conjectural) Galois representation associated to an eigenclass on the Bianchi group, then $r = \wedge^2 (\rho \oplus \rho^c)$ is a six dimensional (reducible) representation which is a direct sum $r = s \oplus \psi \oplus \psi^c$ for a four dimensional representation $s = \rho \otimes \rho^c$ and a Grossencharacter $\psi$ and its conjugate (which are related to the central character of the original form and its conjugate). Now the characteristic polynomials of Frobenius on this Galois representation are, by Eichler–Shimura, the characteristic polynomials of Hecke on the image of this cohomology class in the cohomology $H^*(X_{H})$. Without assuming one has $\rho$, one can phrase the above purely in terms of Satake parameters, but this way of saying it makes clearer what is going on, even though we don’t know yet that $\rho$ actually exists. If one could find the Galois representation $r$ (and in particular $s$) inside the etale cohomology of $X_H$ one would (almost) be done, but instead, the classes which actually turn up in etale cohomology in these degrees are the reducible terms in $r$ corresponding to the Grossencharacters rather than to the interesting representation $s$ we are looking for. So as above, even in characteristic zero, one has the interesting Hecke eigenclass, but not the Galois representation.

These examples suggest that to understand what is going on we first need to get a better understanding of Shimura varieties. Most of the recent history of understanding Shimura varieties (and the Galois representations associated to automorphic forms) has concentrated on the cohomology arising from cuspidal automorphic representations. In this classical setting, the automorphic representations have a classical avatar as global sections of certain coherent bundles on $X_H$. (For example, classical modular forms of weight $\ge 1$ are global sections of the line bundle $\omega^{\otimes k}$.) If we want to restrict to cusp forms, we can also take the corresponding extension of these sheaves to minimal (or toroidal, doesn’t matter) compactifications which vanish appropriately at the boundary. If we denote these automorphic bundles by $\mathcal{E}_{\mathrm{sub}}$, then another way of saying this is that the action of Hecke operators $\mathbf{T}$ on

$\bigoplus_{\mathcal{E}} H^0(\overline{X}_{H},\mathcal{E}_{\mathrm{sub}})$

is now understood if $X_H$ is, for example, a Shimura variety of unitary type over a totally real field. Even getting this far is a somewhat monumental task that required, amongst other things, Ngo’s work on the Fundamental Lemma, work of Kottwitz, Clozel, some large fraction of Jussieu, the work of Shin, and many more. In fact, as far as local-global compatibility goes, the ink is barely dry on the most recent work. Now we can at least state, in vague terms, the following:

Theorem [Scholze, IV.3.1]: For (many) Shimura varieties $X_H$, the action of $\mathbf{T}$ on torsion classes in Betti cohomology factors through the action on coherent cusp forms in characteristic zero.

Two examples: If $X_H$ is the modular curve, then this says that the action of Hecke operators on $H^1(X_H,\mathbf{Z}/p^n \mathbf{Z})$ can be realized by the action on classical modular cuspidal eigenforms modulo powers of $p$. Given how we think about modular forms, this is almost tautological, because, by Eichler-Shimura, we can pass between cohomology classes and classical modular forms (in this case, we can even do this via the Hodge decomposition of $H^1$). However, there is a little wrinkle: we do see Eisenstein classes in Betti cohomology, and this theorem says that we can realize these as coming from cusp forms, so this result also implies that there exist cusp forms which are congruent to Eisenstein classes modulo $p^n$. Since we are ultimately interested in classes coming from the boundary of some compactification, we don’t want to ignore this case. Still, it’s not so difficult to prove.

If $X_H$ comes from $U(2,1)/\mathbf{Q}$ (so it is a arithmetic complex hyperbolic manifold of real dimension $4$, also known as a Picard modular surface), then we can look at the group $H_1(X_H,\mathbf{Z}_p)$. The characteristic zero classes here are known to correspond to endoscopic automorphic representations (and thus to not exist in the co-compact case) and are understood. However, unlike in the modular curve case, we no longer know that this group is torsion free, and in general, it may not be. So, a priori, all we know about the torsion classes and their Hecke operators is that there exists a Galois representation which is annihilated by the characteristic polynomial of $T_p$, using Eichler-Shimura. These polynomials are all of fixed degree (three in this case), but that doesn’t give any lower bound on the dimension of this representation. This is an even more stark example of the well known phenomenon that Eichler-Shimura is pretty much useless for constructing Galois representations outside the case of dimension two where knowing both the trace and determinant tells you a lot. For example, suppose you have an irreducible representation $V$ of a finite group $G$ in characteristic zero such that all the elements of $g$ have a minimal polynomial of degree at most $d$: then you can’t a priori bound the dimension of $V$! As an example, the extra-special $2$-group of order $2^{1 + 2n}$ has a representation of dimension $2^n$ all of whose elements have images satisfying the degree two polynomials $x^2 – 1 = 0$ or $x^2 + 1 = 0$. So, before Scholze, we could not say anything about the dimensions of mod-$p$ Galois representations arising from torsion in the first homology of $U(2,1)$. However, using Scholze, we can now deduce that any such representation comes from a classical cusp form, and hence must (in this case) have dimension three!

Posted in Mathematics | Tagged Galois Representations, Langlands, Peter Scholze, torsion | Leave a comment

Parenthood

En Passant III

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

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

Gerookte paling op de Albert Cuypmarkt

Effective Motives

Scholze on Torsion, Part IV

Scholze on Torsion, Part III

Scholze on Torsion, Part II

Scholze on Torsion, Part I

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