Vol. 10, No. 6, 2016

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Heegner divisors in generalized Jacobians and traces of singular moduli

Jan Hendrik Bruinier and Yingkun Li

Vol. 10 (2016), No. 6, 1277–1300
Abstract

We prove an abstract modularity result for classes of Heegner divisors in the generalized Jacobian of a modular curve associated to a cuspidal modulus. Extending the Gross–Kohnen–Zagier theorem, we prove that the generating series of these classes is a weakly holomorphic modular form of weight $\frac{3}{2}$. Moreover, we show that any harmonic Maass form of weight $0$ defines a functional on the generalized Jacobian. Combining these results, we obtain a unifying framework and new proofs for the Gross–Kohnen–Zagier theorem and Zagier’s modularity of traces of singular moduli, together with new geometric interpretations of the traces with nonpositive index.

Keywords
Singular moduli, generalized Jacobian, Heegner point, Borcherds product, harmonic Maass form
Mathematical Subject Classification 2010
Primary: 14G35
Secondary: 14H40, 11F27, 11F30
Milestones
Received: 27 August 2015
Revised: 20 April 2016
Accepted: 19 May 2016
Published: 30 August 2016
Authors
 Jan Hendrik Bruinier Fachbereich Mathematik Technische Universität Darmstadt Schlossgartenstrasse 7 D-64289 Darmstadt Germany Yingkun Li Fachbereich Mathematik Technische Universität Darmstadt Schlossgartenstrasse 7 D-64289 Darmstadt Germany