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I would like to know what estimates exist for the dimension of $H^d({\rm GL}_2(\mathcal{O}_{K,S}),\mathbb{Q})$ where $\mathcal{O}_{K,S}$ is a ring of $S$-integers in a number field $K$ and $d$ is the virtual cohomological dimension of the group ${\rm GL}_2(\mathcal{O}_{K,S})$. As the title says, I am mostly interested in lower bounds or asymptotic statements measuring the growth of the top cohomology in terms of arithmetic information (like genus, discriminant of $K$ or some such number , and the number of places in $S$). Moreover, information on the possible methods to obtain such estimates (from Euler characteristic computations, via study of Hecke operators, representation theory or so) would be very welcome.

An example of a formula in the spirit of the question is the estimate of the cuspidal cohomology of Bianchi groups in

  • J. Rohlfs. On the cuspidal cohomology of the Bianchi modular groups. Math. Z. 188 (1985), no. 2, 253–269.

I'm wondering if there are similar statements for general rings of $S$-integers. I haven't even been able to find asymptotics for the rational cohomology of ${\rm GL}_2(\mathbb{Z}[1/n])$ for $n\to \infty$ in the literature.

Maybe one more point describing my motivation: in the case of a function field of a curve $C$ over a finite field, the situation is easier to understand. The top rational cohomology can be computed in terms of Gauss-Bonnet formulas and is then related to the zeta-value $\zeta_C(-1)$ for the curve. Estimates for the rational cohomology then follow from the Weil conjectures (so the growth is roughly $q^{2g+s-3}$ with $g$ the genus of the curve, $s$ the number of places, and $q$ the size of the ground field). Essentially I want to know if a similar picture exists (possibly conjecturally?) on the number field side.

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Let $\mathcal{O}$ be the ring of integers in an algebraic number field $k$ and let $\text{cl}(\mathcal{O})$ be the class number of $\mathcal{O}$. In my paper "Integrality in the Steinberg module and the top-dimensional cohomology of $\text{GL}_n(\mathcal{O}_k)$" (joint with Church and Farb), I prove the following two theorems:

  1. In its vcd, the dimension of the rational cohomology group of $\text{SL}_n(\mathcal{O}_k)$ is at least $(\text{cl}(\mathcal{O})-1)^n$. In particular, it is nonzero if $\text{cl}(\mathcal{O}) > 1$.

  2. Conversely, assume that $\text{cl}(\mathcal{O})=1$. Furthermore, assume either that $k$ has a real embedding or that $\mathcal{O}$ is Euclidean. Then in their vcd's the rational cohomology of $\text{SL}_n(\mathcal{O}_k)$ and $\text{GL}_n(\mathcal{O}_k)$ vanishes. There is a similar vanishing result for the homology with certain twisted coefficients.

These are Theorems C and B of that paper (we remark that Theorem C also claims a similar bound for GL, but there is a subtle error in the proof for GL. Our techniques give some lower bound, but we haven't pinned down precisely what it is. That should be corrected soon.).

I remark that in the Euclidean case the vanishing result in 2 above is an old theorem of Lee and Szczarba from

R. Lee and R. H. Szczarba, On the homology and cohomology of congruence subgroups, Invent. Math. 33 (1976), no. 1, 15-53.

For $n=2$, the lower bound in 1 above is pretty classical, but the vanishing result in 2 seems to be new (at least in the non-Euclidean case). I'm not sure what happens for $S$-arithmetic groups.

The paper can be downloaded from my webpage here.

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  • $\begingroup$ Thanks, that's very interesting. If you have any intuition as to how sharp your lower bounds are or how well they reflect the "growth" of the top rational cohomology, I would be interested to know. In the function field case, the contribution from the zeta function is more significant than contributions from class groups. But then, the asymptotics are exponential with base $q$ (the size of the base field). So the $\mathbb{F}_1$-philosophy of specializing to $q=1$ doesn't yield anything. $\endgroup$ – Matthias Wendt Dec 20 '17 at 17:24
  • $\begingroup$ @MatthiasWendt: Growth with respect to what? I don't have a good sense as to how sharp our lower bounds are (though as far as I know they're the only known cohomology in the vcd, at least for n large). The one indication that they might be really capturing something is the vanishing result I listed, which says that in favorable situations if they don't give any cohomology, then there is no cohomology to be found. I think of the vanishing result as the main theorem of the paper I listed. The lower bound result was really there to show that the hypothesis of class number $1$ was needed. $\endgroup$ – Andy Putman Dec 20 '17 at 20:00
  • $\begingroup$ One speculation I might have (based on lots of attempts to try to find other cohomology classes) is that our lower bound might be sharp if the class group is the direct sum of copies of $\mathbb{Z}/2$, but is probably not sharp otherwise. But I don't know how to prove this. $\endgroup$ – Andy Putman Dec 20 '17 at 20:01

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