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Let $K$ be a number field, which can be $\mathbb Q(\zeta)$ with $\zeta$ a root of unity if that helps, and let $\operatorname{SL}_n(\mathbb{Z})$ act on $(K^\times)^n:=K^\times\times\cdots\times K^\times$ by «matrix multplication», so that $M\cdot (\lambda_1,\dots,\lambda_n)=(\lambda_1^{m_{1,1}}\cdots\lambda_n^{m_{1,n}},\dots, \lambda_1^{m_{n,1}}\cdots\lambda_n^{m_{n,n}})$.

Is the cohomology $H^*(\operatorname{SL}_n(\mathbb{Z}),(K^\times)^n)$ known, at least in degree $2$?

If $\mu$ is the group of roots of unity in $K$, then $\mu^n=\mu\times\cdots\times\mu$ is a $\operatorname{SL}_n(\mathbb{Z})$-submodule of $(K^\times)^n$, and one can also ask

Is the cohomology $H^*(\operatorname{SL}_n(\mathbb{Z}),\mu^n)$ known, at least in degree $2$?

The two cohomologies are related by a long exact sequence, of course.

I am really interested in the case where $n=2$ which should be easier (?)...

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    $\begingroup$ Can't you exploit that SL_2(Z) is the amalgam of C_4 and C_6 along C_2 and apply the Mayer-Vietoris sequence? $\endgroup$
    – tj_
    Commented Aug 6, 2021 at 19:56
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    $\begingroup$ That does work great if the coefficients have trivial action. You find, for example, that $H^2(\operatorname{SL}_2(\mathbb Z),\mathbb Z)$ is $\mathbb Z/12\mathbb Z$ and $0$ in odd degrees, so you can compute $H^2$ with values in finite cyclic groups with trivial action. $\endgroup$
    – Mariano Suárez-Álvarez
    Commented Aug 6, 2021 at 20:00
  • $\begingroup$ Whatever the answer is, I would not think it would have much to do with $K$. What is known about the cohomology in low degrees of $SL_n(\mathbb Z)$ with $\mathbb Z^n$ coefficients? $\endgroup$
    – Tom Goodwillie
    Commented Aug 6, 2021 at 20:22
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    $\begingroup$ The Mayer-Vietoris sequence holds for all coefficients (cf. Brown VII.9) and the cohomology of cyclic groups is easy. So I would guess that you can get some information from MV. $\endgroup$
    – tj_
    Commented Aug 6, 2021 at 20:25
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    $\begingroup$ Perhaps the following reference can be useful: "The cohomology of the braid group B_3 and of SL_2(Z) with coefficients in a geometric representation", Filippo Callegaro, Fred Cohen, Mario Salvetti. $\endgroup$
    – Dan Petersen
    Commented Aug 7, 2021 at 7:54

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