Funke, Jens and Millson, John (2014) 'The geometric theta correspondence for Hilbert modular surfaces.', Duke mathematical journal., 163 (1). pp. 65-116.
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 Hirzebuch’s smooth complex analytic compactification of the Hilbert modular surface with the (real) Borel-Serre compactification. The various algebro-geometric quantities that occur in  are replaced by topological quantities associated to 4-manifolds with boundary. In particular, the “boundary contribution” in  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.
|Full text:||(AM) Accepted Manuscript|
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|Publisher Web site:||http://dx.doi.org/10.1215/00127094-2405279|
|Date accepted:||07 April 2013|
|Date deposited:||21 December 2013|
|Date of first online publication:||08 January 2014|
|Date first made open access:||08 January 2014|
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