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If I understood my automorphic forms correctly, at least cusp forms can be thought of as elements of $L^2(G/\Gamma)$ for a $G = \text{SL}_2(\mathbb{R})$ and $\Gamma = \text{SL}_2(\mathbb{Z})$ or a suitable congrence subgroup.

I want to know if there is an analogous result for hyperbolic-3 space modulo a Bianchi group. For example: $$ \mathbb{H}^3 \,/\, \text{PSL}_2 \big(\mathbb{Z}[i] \big) $$

This is a group action by isometries. They inherit from this action of the full group:

\begin{eqnarray*} \text{SL}_2(\mathbb{C}) \times \mathbb{H}^3 &=& \mathbb{H}^3\\ (g,z) &=& \frac{az+b}{cz+d} \end{eqnarray*}

In this case, is there a version of modular forms (or cusp forms) here? I have never seen a discussion of modular forms over quaternionic arguments. I suppose the method of images could produce an invariant function:

$$ f(z) = \sum_{\gamma \in \text{PSL}_2(\mathbb{Z}[i])} f_0( \gamma \, z) $$

I am hoping or something more explicit. Is there an analog of theta function or newform in this setting? My guess this should be an element of the space $L^2(G/\Gamma)$ that I have constructed.

$$ f_0(a+bi+cj+dk) = e^{2\pi i \, (a^2 + b^2+c^2 + d^2)} = q^{z \overline{z}} $$

This is just a speculation. Certainly this will be invariant but it won't be in $L^2$.

$\text{PSL}_2(\mathbb{Z}[i])$ is a textbook exmple of a Kleininan group and there are basica ways to construct hyperbolic 3-manifolds, such as $\mathbb{H}^3/\text{PSL}_2(\mathbb{Z}[i])$.

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In the case of $\mathbb{H}/\text{SL}_2(\mathbb{Z})$ we could construct modular forms in fairly using Eisenstein series or theta functions. And these examples will be fairly explicit since we often know their Fourier coefficients. And I'm sure the question for modular forms over Bianchi groups is discussed somewhere $\text{PSL}_2(\mathbb{Z}[i])$

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    $\begingroup$ I'm not sure I understand the hyperbolic three space you are using to write down the equations $\endgroup$
    – Watson Ladd
    Dec 5, 2017 at 2:02
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    $\begingroup$ Have you tried looking at Elstrodt-Grunewald-Mennicke? $\endgroup$
    – Kimball
    Dec 5, 2017 at 3:08
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    $\begingroup$ The hyperbolic three space is the symmetric space attached to $PSL_3({\mathbb C})$. So this is just the usual notion of automorphic forms for linear groups. $\endgroup$
    – user1688
    Dec 5, 2017 at 8:48
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    $\begingroup$ @johnmangual The hyperbolic $3$ space can be realized as an upper half plane model by a subset of Hamiltonian quaternions having zero real part: $$\{P:=z+rj\mid z\in \mathbb{C}, r\in \mathbb{R}^+\}.$$ $SL_2(\mathbb{C})$ acts transitively by generalized Mobius transformation: $$\begin{pmatrix}a&b\\c&d\end{pmatrix}P=(aP+b)(cP+d)^{-1}.$$ As quaternions are not commutative $\frac{aP+b}{cP+d}$ does not make sense. Then the space $$SL_2(\mathbb{Z}[i])\backslash \mathbb{H}^3 = SL_2(\mathbb{Z}[i])\backslash SL_2(\mathbb{C})/SU(2).$$ $\endgroup$
    – Subhajit Jana
    Dec 5, 2017 at 11:40
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    $\begingroup$ ....This modular domain has one cusp, which can be checked via strong approximation. You can talk about the Maass forms on this space, for example, a real analytic Eisenstein series would be $$E_s(P):=\sum_{\gamma\in\Gamma_\infty\backslash\Gamma}r(\gamma P)^s.$$ $\endgroup$
    – Subhajit Jana
    Dec 5, 2017 at 11:41

2 Answers 2

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In addition to @Kimball's good suggestion of Elstrodt-Grunewald-Mennicke, chapter one of my book (available on-line at http://www.math.umn.edu/~garrett/m/v/Book_28Aug2017.pdf) treats the example of $SL(2,\mathbb Z[i])\subset SL(2,\mathbb C)\sim SO(3,1)$ in some detail (as well as the iconic $SL(2,\mathbb Z)$, and also $Sp^*(1,1)(\mathfrak o)\subset Sp^*(1,1)(\mathbb R) \sim SO(4,1)$ and $SL(2,\mathfrak o)\subset SL(2,\mathbb H)\sim SO(5,1)$, where $\mathfrak o$ is the ring of Hurwitz integers in $\mathbb H$. The unicuspidality is proven, pseudo-Eisenstein series are spectrally decomposed in terms of Eisenstein series, and so on.

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  • $\begingroup$ +1 something like this. There are a number of books on 3-manifolds such as Voight's book on quaternions or Marden's book "Outer Circles" but those don't deal with modular forms . The automorphic forms framework always tells us something exists, but their discussions are very abstract and very technical. My reasoning was $\mathbb{Z}[i]$ is a Euclidean domain and therefore $\text{PSL}_2(\mathbb{Z}[i])$ should be somewhat nice . Therefore it should be easy to build modular forms on $\mathbb{H}^3/\text{PSL}_2(\mathbb{Z}[i])$ by analogy to the $\mathbb{H}/\text{SL}_2(\mathbb{Z})$ case. $\endgroup$
    – john mangual
    Dec 5, 2017 at 17:40
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Please have a look at "Period integrals of cohomology classes which are represented by Eisenstein Series" by Guenter Harder, it appeared in a proceedings of a conference in Bombay. He only deals with the case of $PSL(2, \mathbb{Z}[i])$. Its a beautiful paper, and it will take you through the wonderful world of Eisenstein Cohomology in this particular case.
After this you can also look at "Eisenstein series for $SL_2(\mathbb{Z}[i])$ and special values of L-functions" by Guenter Harder. Its a sequel to the paper mentioned above and gets more into the arithmetic aspects of the L-series associated with Eisenstein modular forms. Again, explicit examples are included.

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