Let \(F\) be a number field, and let \(\mathbf{G}\) be a reductive group over \(F\), and let \(\Gamma\) be a congruence subgroup of \(\mathbf{G}(\mathcal{O}_F)\). I can hear BC objecting that this doesn’t make sense without extra choices; if you have such an objection, please make such choices. Matt and I have made various conjectures concerning the vanishing of the completed cohomology groups \(\widetilde{H}^{n}\) in the range \(n > q_0\), where \(q_0\) has been defined for all time by Borel and Wallach. (And what is \(q\), you ask? Well, having just consulted [BW] by downloading a pirated djvu copy, I can tell you that \(2q = \mathrm{dim}(G/K)\) [4.3, p.67]. What’s that, you say — \(q\) isn’t even always an integer? Nope!) Several cases of this conjecture were proved by Peter (in particular, in the Shimura variety context), but the general conjecture seems quite hard (not that the Shimura variety case was a cakewalk!). For example, when \(G = \mathrm{GL}(1)\), then \(q_0 = 0\) and the conjecture is equivalent to Leopoldt’s conjecture. To remind you, one way of stating Leopoldt’s conjecture is that the profinite topology on \(\mathcal{O}^{\times}_F \times \mathbf{Z}_p\) coincides with the topology coming from the \(p\)-adic topology — that is, units are close if they are close modulo powers of \(p\). In contrast, one can ask for the weaker statement that that the profinite topology on \(\mathcal{O}^{\times}_F \times \mathbf{Z}_p\) coincides with the congruence topology, namely, the topology coming from looking at units modulo \(N\) for any ideal \(N\). This turns out to be unconditionally true and not too difficult, although it is not quite as obvious as it may seem (the same can be said of LC). It motivates, however, the following conjecture:
Conjecture (Horizontal Vanishing) Let \(n > q_0\). Then the following direct limit vanishes
\(\displaystyle{\lim_{K} H^n(X(K),\mathbf{F}_p) = 0}\)
as \(K\) ranges over all compact open subgroups of \(\mathbf{G}(\mathbf{A}^f_F)\).
There is an equivalent formulation of this conjecture in terms of group cohomology for arithmetic lattices. Because the conjecture is known for \(\mathrm{GL}(1)\), one can also pass easily enough between \(\mathrm{SL}\) and \(\mathrm{GL}\). For example, for \(\mathrm{SL}_N(\mathbf{Z})\) and \(n > 2\) it has the following formulation: Any cohomology class in \(H^n(\mathrm{SL}_N(\mathbf{Z}),M)\) for a finite discrete module \(M\) capitulates in some congruence subgroup, providing that
\(\displaystyle{n > \left\lfloor \frac{N^2}{4} \right\rfloor}.\)
This vanishing is related to the concept of virtual cohomological dimension. The virtual cohomological dimension of a group \(G\) is the smallest integer \(m\) such that there exists a finite index subgroup \(H \subset G\) such that every cohomology class in degree \(> m\) capitulates in \(H\). The notion being considered here is what one gets by reversing the quantifiers — one only insists that the classes capitulate in smaller and smaller groups (in addition, we insist that \(H\) is a congruence subgroup, although that is not too restrictive when the rank is \(\ge 2).\) There is a trivial bound \(m \ge q_0\), but this bound is not at all sharp. Since this seems an a priori interesting notion, let’s define it:
Definition: pro-virtual cohomological dimension: Let \(G\) be a group, and let \(p\) be a prime. Say that \(\mathrm{pvcd}_p(G) < m\) if, for every discrete \(G\)-module \(M\) annihilated by \(p\), and every cohomology class \([c] \in H^n(G,M)\) for some \(n \ge m\), there exists a finite subgroup \(H\) so that the restriction of \([c]\) to \(H^n(H,M)\) vanishes. Say that \(G\) has pro-virtual cohomological dimension \(< m\) if \(\mathrm{pvcd}_p(G) < m\) for all \(p\).
I have nothing profound to say about whether this concept is relevant beyond the example at hand. As you can see for \(\mathrm{SL}_N(\mathbf{Z})\), the virtual cohomological dimension and pro-virtual cohomological dimension are conjecturally quite different, the latter being given conjecturally by the formula above (at least when \(N > 2\)), and the former by \(\displaystyle{\binom{N}{2}}.\)
I wanted to remark in this post that the Horizontal Vanishing conjecture is, at least after localization at a non-Eisenstein maximal ideal \(\mathfrak{m}\), a consequence of modularity lifting results (in the spirit of [CG]). Namely, the entire point of that method is that the patched complex has cohomology concentrated in a single degree (\(q_0\)), which amounts to saying that cohomology classes in \(H^{q_0 + i}(X(K),\mathbf{F}_p)\) can be annihilated after passing to some auxiliary level coming from some choice of Taylor-Wiles primes. Now many aspects of this argument are still conditional (note that to annihilate classes of deep \(p\)-power level, one would need corresponding local-global compatibility relating Galois representations associated to torsion classes to quotients of Kisin deformation rings, at the very least), but perhaps it is a less hopeless task than trying to prove Leopoldt’s conjecture.
It’s instructive to consider what is possibly the simplest case of this conjecture beyond Shimura varieties, namely, \(\mathrm{GL}(2)\) over an imaginary quadratic field (here \(q_0 = 1\), so the claim is that one can kill classes in \(H^2).\) Here at least one doesn’t have to worry about vanishing of cohomology outside the indicated range. Local-global compatibility is still a problem, but one possibly way to get around this is to work at all p-power levels at once, namely, to patch the completed cohomology groups. (Matt, Toby, and I chatted over roast duck at Sun Wah what patching completed cohomology for general groups should look like.) Since one certainly has Galois representations, one gets “for free” the fact that the patched modules are modules over the appropriate power series ring of the local deformation ring. On the other hand, as Matt cautioned at the ‘Pig, it is no longer so easy to do naive arguments with codimensions, because the patched objects are not finitely generated over the ring of diamond operators, but only over a non-commutative group algebra, which leads into questions relating to the size of the corresponding \(p\)-adic representations, which leads back to questions concerning local-global compatibility in \(p\)-adic local Langlands.
I wonder, however, if there are any softer arguments in any special cases.
This pro-vanishing condition is almost in the literature. In the exercises about “good” groups (Galois Cohomology I 2.6), Serre says that such vanishing in all positive degrees is equivalent to being “good.” In particular, Bianchi groups are “good” so they satisfy this vanishing condition in all degrees. However, that is for the full profinite topology, while you want the congruence topology. So the congruence vanishing property must fail. But it must already fail in dimension 1, so that leaves the question of dimension 2 open.
The difference between Serre’s property and your property is that he considers a range of low degrees, from zero to an upper bound, while you consider a range of high degrees, from a lower bound on up.
On another note, I was a bit disoriented by your reuse of the variable q. Of course people use it all the time to denote degree in cohomology, and thus reuse it, but you used it for two closely related degrees. The second time was the vcd of an abstract group. The first time was the Borel-Wallach q, which is half of (a bound on) the vcd of a lattice.
Thanks, I changed the potentially confusing use of $latex q$ to $latex m$ instead.
I do indeed know about Serre’s notion of “goodness,” although, as you say, it’s somewhat orthogonal to what is being considered here. The “interesting” cohomology of arithmetic groups is precisely the cohomology that is *not* coming from the “local” (in the number theory sense) cohomology of the congruence completion, so the general concept of goodness is in some sense opposite to what one wants in this case.