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arXiv · 1508.07112

Heegner divisors in generalized Jacobians and traces of singular moduli

Jan Hendrik Bruinier·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 forms 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 non-positive index.

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Jan Hendrik Bruinier, Yingkun Li. 2017-02-21. Heegner divisors in generalized Jacobians and traces of singular moduli. https://doi.org/10.2140/ant.2016.10.1277

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