Algebra & Number Theory

Heegner divisors in generalized Jacobians and traces of singular moduli

Jan Hendrik Bruinier and Yingkun Li

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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

Subjects
Primary: 14G35: Modular and Shimura varieties [See also 11F41, 11F46, 11G18]
Secondary: 14H40: Jacobians, Prym varieties [See also 32G20] 11F27: Theta series; Weil representation; theta correspondences 11F30: Fourier coefficients of automorphic forms

Keywords
Singular moduli generalized Jacobian Heegner point Borcherds product harmonic Maass form

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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