Scholze on Torsion, Part IV

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:

1. 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 \(m\).
2. 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:

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

1. Understand the tower (either the Lubin-Tate tower or the corresponding tower of modular curves) as an actual geometric object \(\mathcal{X}\) (perfectoid space).
2. Construct a period map \(\pi: \mathcal{X} \rightarrow \mathbf{P}^1\) (or \(\mathscr{Fl}\)) using \(p\)-adic Hodge theory.
3. 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 \(\mathbf{P}^1\).
4. 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 \(H^*(\mathcal{X}^*,\mathcal{E})\)?