
Mathematics 2011
The Geometric Theta Correspondence for Hilbert Modular SurfacesAbstract: In a series of papers we have been studying the geometric theta correspondence for noncompact 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 noncompact locally symmetric spaces of orthogonal and unitary groups which generalizes the theory of KudlaMillson in the compact case. 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 that the generating function for the intersection numbers of the HirzebruchZagier cycles is a classical modular form of weight 2. In our approach we replace Hirzebuch's smooth complex analytic compactification $\tilde{X}$ of the Hilbert modular surface $X$ with the (real) BorelSerre compactification $\bar{X}$. The various algebrogeometric quantities are then replaced by topological quantities associated to 4manifolds with boundary. In particular, the "boundary contribution" in HirzebruchZagier is replaced by sums of linking numbers of circles (the boundaries of the cycles) in the 3manifolds of type Sol (torus bundle over a circle) which comprise the BorelSerre boundary.
