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The geometric theta correspondence for Hilbert modular surfaces

Funke, Jens; Millson, John

The geometric theta correspondence for Hilbert modular surfaces Thumbnail


John Millson


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.

Journal Article Type Article
Acceptance Date Apr 7, 2013
Online Publication Date Jan 8, 2014
Publication Date Jan 1, 2014
Deposit Date Mar 19, 2012
Publicly Available Date Jan 8, 2014
Journal Duke Mathematical Journal
Print ISSN 0012-7094
Electronic ISSN 1547-7398
Publisher Duke University Press
Peer Reviewed Peer Reviewed
Volume 163
Issue 1
Pages 65-116
Public URL


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