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arXiv · 1609.08100

On Divisors of Modular Forms

Kathrin Bringmann·Ben Kane·Steffen Löbrich·Ken Ono·Larry Rolen
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Abstract

The denominator formula for the Monster Lie algebra is the product expansion for the modular function $j(z)-j(\tau)$ given in terms of the Hecke system of $\operatorname{SL}_2(\mathbb Z)$-modular functions $j_n(\tau)$. It is prominent in Zagier's seminal paper on traces of singular moduli, and in the Duncan-Frenkel work on Moonshine. The formula is equivalent to the description of the generating function for the $j_n(z)$ as a weight 2 modular form with a pole at $z$. Although these results rely on the fact that $X_0(1)$ has genus 0, here we obtain a generalization, framed in terms of polar harmonic Maass forms, for all of the $X_0(N)$ modular curves. We use these functions to study divisors of modular forms.

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Kathrin Bringmann, Ben Kane, Steffen Löbrich, Ken Ono, Larry Rolen. 2016-09-26. On Divisors of Modular Forms. https://arxiv.org/abs/1609.08100

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