Number theoretic generalization of the monster denominator formula

作     者：Bringmann, Kathrin Kane, Ben Löbrich, Steffen Ono, Ken Rolen, Larry 

作者机构：Mathematical Institute University of Cologne Weyertal 86-90 Cologne50931 Germany Department of Mathematics University of Hong Kong Pokfluam Hong Kong Department of Mathematics and Computer Science Emory University AtlantaGA30022 United States Hamilton Mathematics Institute School of Mathematics Trinity College Dublin 2 Ireland 

出 版 物：《arXiv》 (arXiv)

年 卷 期：2017年

核心收录：

摘      要：The denominator formula for the Monster Lie algebra is the product expansion for the modular function j(z) − j(τ) in terms of the Hecke system of SL2(Z)-modular functions jn(τ). This formula can be reformulated entirely number theoretically. Namely, it is equivalent to the description of the generating function for the jn(z) as a weight 2 modular form in τ with a pole at z. Although these results rely on the fact that X0(1) has genus 0, here we obtain a generalization, framed in terms of polar harmonic Maass forms, for all of the X0(N) modular curves. In this survey of recent work, we discuss this generalization, and we offer an introduction to the theory of polar harmonic Maass forms. We conclude with applications to formulas of Ramanujan and Green s functions. Copyright © 2017, The Authors. All rights reserved.