## Algebra & Number Theory

### Heegner divisors in generalized Jacobians and traces of singular moduli

#### 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 $3 2$. 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.

#### Article information

Source
Algebra Number Theory, Volume 10, Number 6 (2016), 1277-1300.

Dates
Received: 27 August 2015
Revised: 20 April 2016
Accepted: 19 May 2016
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1510842553

Digital Object Identifier
doi:10.2140/ant.2016.10.1277

Mathematical Reviews number (MathSciNet)
MR3544297

Zentralblatt MATH identifier
1348.14072

#### Citation

Bruinier, Jan Hendrik; Li, Yingkun. Heegner divisors in generalized Jacobians and traces of singular moduli. Algebra Number Theory 10 (2016), no. 6, 1277--1300. doi:10.2140/ant.2016.10.1277. https://projecteuclid.org/euclid.ant/1510842553

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