Heegner divisors in generalized Jacobians and traces of singular moduli

Bruinier, Jan Hendrik; Li, Yingkun

doi:10.2140/ant.2016.10.1277

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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

DOI: 10.2140/ant.2016.10.1277

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

Vol. 10, No. 6, 2016