Maass forms on PSL(2,Z[i]) (awaiting review)

A Maass form is a smooth, square-integrable, automorphic eigenfunction of the Laplacian, $$f\in C^\infty({\mathcal H}),\quad f\in L^2(\Gamma\backslash{\mathcal H}),\quad f(\gamma z)=f(z)\ \forall\gamma\in\Gamma,\quad (\Delta+\lambda)f(z)=0.$$ Maass forms on PSL$(2,{\mathbb Z}[i])$ live in the upper half-space ${\mathcal H}=\{z=x+iy \mid x\in{\mathbb C},\ y>0\}$, equipped with the hyperbolic metric $ds=|dz|/y$.

The Picard group $\Gamma=$PSL$(2,{\mathbb Z}[i])$ is a discrete, cofinite, but non-cocompact subgroup of the group of isometries PSL$(2,{\mathbb C})$ ,where the isometries are given by linear fractional transformations $z\to(az+b)(cz+d)^{-1}$ with $a,b,c,d\in {\mathbb C}$, $ad-bc\not=0$.

Keywords: #Maass #Picard