The geometric theta correspondence for Hilbert modular surfaces

Funke, Jens; Millson, John

Abstract

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.

Citation

Funke, J., & Millson, J. (2014). The geometric theta correspondence for Hilbert modular surfaces. Duke Mathematical Journal, 163(1), 65-116. https://doi.org/10.1215/00127094-2405279

• 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
• DOI: https://doi.org/10.1215/00127094-2405279