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On certain cohomology sets attached to Riemann surfaces. (English) Zbl 0967.11020

For \(N\geq 3\) let \(\Gamma(N)\) be the principal congruence subgroup of \(PGL_2(\mathbb{Z})\) of level \(N\). Let \(\Sigma\) be the subgroup generated by \(\left( \begin{smallmatrix} 0 &-1\\ 1 &0\end{smallmatrix} \right)\). Then \(\Sigma\) normalizes \(\Gamma(N)\), so the first nonabelian cohomology set \(\text{H}^1 (\Sigma, \Gamma(N))\) is defined. In the present paper it is computed in an explicit fashion that \[ \# \text{H}^1(\Sigma,\Gamma(N))= \begin{cases} \frac 14 \# SO_2 (\mathbb{Z}/N \mathbb{Z}) &\text{if }N\equiv 0\pmod 4,\\ \frac 12 \# SO_2 (\mathbb{Z}/N\mathbb{Z}) &\text{otherwise}. \end{cases} \] {}.

MSC:

11F75 Cohomology of arithmetic groups
30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
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