The Geometric Theta Correspondence for Hilbert Modular Surfaces

TL;DR

This paper develops a geometric theta correspondence for Hilbert modular surfaces, providing a new proof and extension of a classical modular form theorem using topological methods and Borel-Serre compactification.

Contribution

It introduces a topological approach to the geometric theta lift for Hilbert modular surfaces, extending the Hirzebruch-Zagier theorem to all finite index subgroups.

Findings

Proves the generating function for intersection numbers is a modular form of weight 2.

Replaces complex analytic compactification with Borel-Serre compactification.

Expresses boundary contributions as linking numbers in 3-manifolds.

Abstract

In a series of papers we have been studying the geometric theta correspondence 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. 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 Hirzebruch-Zagier 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) Borel-Serre compactification $\bar{X}$. The various algebro-geometric quantities are then replaced by topological quantities associated to 4-manifolds with boundary. In particular, the "boundary contribution" in Hirzebruch-Zagier 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.