Duke Mathematical Journal

Borcherds products and arithmetic intersection theory on Hilbert modular surfaces

Jan H. Bruinier, José I. Burgos Gil, and Ulf Kühn

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Abstract

We prove an arithmetic version of a theorem of Hirzebruch and Zagier saying that Hirzebruch-Zagier divisors on a Hilbert modular surface are the coefficients of an elliptic modular form of weight 2. Moreover, we determine the arithmetic self-intersection number of the line bundle of modular forms equipped with its Petersson metric on a regular model of a Hilbert modular surface, and we study Faltings heights of arithmetic Hirzebruch-Zagier divisors

Article information

Source
Duke Math. J., Volume 139, Number 1 (2007), 1-88.

Dates
First available in Project Euclid: 13 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1184341238

Digital Object Identifier
doi:10.1215/S0012-7094-07-13911-5

Mathematical Reviews number (MathSciNet)
MR2322676

Zentralblatt MATH identifier
1208.11077

Subjects
Primary: 11G18: Arithmetic aspects of modular and Shimura varieties [See also 14G35]
Secondary: 11F41: Automorphic forms on GL(2); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces [See also 14J20] 14C17: Intersection theory, characteristic classes, intersection multiplicities [See also 13H15] 14G40: Arithmetic varieties and schemes; Arakelov theory; heights [See also 11G50, 37P30] 14C20: Divisors, linear systems, invertible sheaves

Citation

Bruinier, Jan H.; Burgos Gil, José I.; Kühn, Ulf. Borcherds products and arithmetic intersection theory on Hilbert modular surfaces. Duke Math. J. 139 (2007), no. 1, 1--88. doi:10.1215/S0012-7094-07-13911-5. https://projecteuclid.org/euclid.dmj/1184341238


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