The geometric theta correspondence for Hilbert modular surfaces

Abstract

We give a new proof and an extension of the celebrated theorem of Hirzebruch and Zagier that the generating function for the intersection numbers of the Hirzebruch–Zagier cycles in (certain) Hilbert modular surfaces is a classical modular form of weight $2$. In our approach, we replace Hirzebruch’s smooth complex analytic compactification of the Hilbert modular surface with the (real) Borel–Serre compactification. The various algebro-geometric quantities that occur in the theorem are replaced by topological quantities associated to $4$-manifolds with boundary. In particular, the “boundary contribution” in the theorem is replaced by sums of linking numbers of circles (the boundaries of the cycles) in $3$-manifolds of type Sol (torus bundle over a circle) which comprise the Borel–Serre boundary.