# spectral decomposition for elliptic surfaces?

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.

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This looks like a Jacobi group. Are you aware of Berndt and Schmidt's book "Element of the Representation theory of the Jacobi group"? Google books has part of chapter 4, which studies what looks like your L$^2$(S). Even if it isn't exactly what you want, if you understand the L$^2$(Y) case, you should be able to analogize their construction to your case. –  B R Nov 10 '10 at 20:48
it seems that the elliptic surface is fibered over the modular curve by compact tori, and thus the new contribution to the spectrum decomposition should appear only in the discrete part, although I know nothing about the concrete formula. And for the Kuga varieties over compact Shimura curves, then still there is only a discrete spectrum. –  turtle Nov 11 '10 at 8:11
thanks! the book is very useful! and also for kuga varieties over compact Shimura curves I think Lee Min Ho has written quite a lot around this topic. –  genshin Nov 11 '10 at 10:23