Scholze on Torsion, Part II

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:

1. You can glue determinants: \(D \rightarrow A/I\) and \(D \rightarrow A/J\) which agree on \(A/(I + J)\) to get a determinant \(D \rightarrow A/(I \cap J).\)
2. 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:

1. 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.)

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2. 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).

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3. 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.

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

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5. 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.

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(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.