batchison.tex

\section{Introduction}

In a series of papers \cite{FM1,FMcoeff,FMres,FMspec} we have been studying the geometric theta correspondence (see below) for non-compact arithmetic quotients of symmetric spaces associated to orthogonal groups. It is our overall goal to develop a general theory of geometric theta liftings in the context of the real differential geometry/topology of non-compact locally symmetric spaces of orthogonal and unitary groups which generalizes the theory of Kudla-Millson in the compact case, see \cite{KM90}. 

In this paper we study in detail the geometric theta lift for \\
Hilbert modular surfaces. In particular, we will give a new proof \\
and an extension (to all finite index subgroups of the Hilbert \\
modular group) of the celebrated theorem of Hirzebruch and Zagier\\
\cite{HZ} that the generating function for the intersection numbers\\
of the Hirzebruch-Zagier cycles is a classical modular form of \\
weight $2$.\footnote{Eichler, \cite{HZ} p.104, proposed a proof using \\
``Siegel's work on indefinite theta functions''. This is what our \\
proof is, though with perhaps more differential geometry than Eichler \\
had in mind.} In our approach we replace Hirzebuch's smooth complex \\
analytic compactification $\tilde{X}$ of the Hilbert modular surface \\
$X$ with the (real) Borel-Serre compactification $\overline{X}$. \\
The various algebro-geometric quantities that occur in \cite{HZ} \\
are then replaced by topological quantities associated to $4$-manifolds \\
with boundary. In particular, the ``boundary contribution'' in \\
\cite{HZ} is replaced by sums of linking numbers of circles \\
(the boundaries of the cycles) in the $3$-manifolds of type \\
Sol (torus bundle over a circle) which comprise the Borel-Serre boundary.\\