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I am looking for a reference for Eisenstein series for discrete subgroups of $SL(2,\mathbb C)$, in particular, finite index subgroups of $SL(2,\mathcal O_K)$ where $K$ is an imaginary quadratic field.

Much work has been done over discrete subgroups of $SL(2,\mathbb R)$, and similarly Eisenstein series on $SL(2,\mathcal O_K)$ itself, but I have not been able to locate this particular case.

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I do not recall off-hand whether J. Elstrodt, E. Grunewald, J. Mennicke, Eisenstein series on three-dimensional hyperbolic spaces and imaginary quadratic fields, J. reine und angew. Math. 360 (1985), 160–213 treats more general cases, but this is a standard reference, anyway.

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    $\begingroup$ This does not answer the question. The paper you mention seems only to treat the full Bianchi modular group $SL_2(O_K)$; and the poster specifically points out that he is aware of references for the full Bianchi modular group, but wants for something treating more general finite-index subgroups of such groups. $\endgroup$
    – David Loeffler
    Oct 5, 2018 at 6:30
  • $\begingroup$ Thanks! I am aware of the reference (and their book also), but most authors I've found have only been working over the full $SL_2(\mathcal O_K)$.. $\endgroup$
    – Tian An
    Oct 5, 2018 at 14:39
  • $\begingroup$ You can find some results for Eisenstein series for $\Gamma=\Gamma_0(q)$ and $K=\mathbb{Q}(i)$ in Qi's paper arxiv.org/pdf/1805.06026.pdf for example. He also has other papers on $SL(2,\mathbb{C})$ where you may find something you want. $\endgroup$
    – BH NT
    Mar 15, 2019 at 8:56

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