I'm looking for explicit formulae for the spectral decomposition of $L^2(S)$, where $S$ is an elliptic surface (of complex dimension 2).

To be precise, the elliptic surface I'm looking at is the complex algebraic surface underlying the universal elliptic curve over a modular curve, which is among the first examples of Kuga varieties. It is expected to be similar to the case of modular curves, as follows:

For modular curves $Y=Y(\Gamma)$ ($\Gamma$ being some congruence subgroup of $SL_2(\mathbb{Z})$), and more generally quotients of hyperbolic upper half plane by a congruence subgroup from some quaternion algebra (namely compact Shimura curves), the spectral decomposition of $L^2(Y)$, namely expansion of a function into a "sum" of eigen-function of the Laplacian of $Y$ (deduced from the hyperbolic metric of the upper half plane), consists of two parts: the discrete spectrum and the continuous spectrum. The discrete spectrum is a Hilbert space with a countable basis consisting of eigenfunctions invariant under the congruence subgroup, and the continuous spectrum is an inseparable Hilbert space, with a basis represented by Eisenstein series (depending on a continuous parameter). Note that if the congruence subgroup comes from a non-split quaternion algebra, then $Y$ is compact and there is no continuous spectrum in $L^2(Y)$.

I wonder if the same thing holds for the elliptic surface, namely the universal elliptic curve over a modular curve. In this case the underlying manifold looks like $$S=\mathbb{Z}^2\rtimes SL_2(\mathbb{Z})\backslash \mathbb{R}^2\times \mathcal{H}$$ It is a locally symmetric hermitian manifold, but no longer of non-compact type, as it contains an Euclidean factor. It still carries a metric with finite volume. I would like to find references, or methods, to decompose $L^2(S)$ along eigen-functions of the Laplacian, with the continuous spectrum given by certain Eisenstein inegrals, and the discrete part given by suitable automorphic, namely functions invariant under the group $\mathbb{Z}^2\rtimes SL_2(\mathbb{Z})$ with tempered growth at infinity.

I don't know if similar situations have been also considered for Kuga varieties, like universal abelian varieties over Shimura curves etc.

Thanks a lot and please fell free to correct the inaccuracies above.