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The geometric theta correspondence for Hilbert modular surfaces.

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 [17] 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 [17] are replaced by topological quantities associated to 4-manifolds with boundary. In particular, the “boundary contribution” in [17] 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.

Item Type:Article
Full text:(AM) Accepted Manuscript
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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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