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On the vanishing of the coefficients of CM eta quotients

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  18 October 2023

Tim Huber ,
Chang Liu ,
James McLaughlin ,
Dongxi Ye ,
Miaodan Yuan  and
Sumeng Zhang
Tim Huber
Affiliation:
School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, Edinburg, TX, USA (timothy.huber@utrgv.edu)
Chang Liu
Affiliation:
School of Mathematics (Zhuhai), Sun Yat-Sen University, Zhuhai, Guangdong, People’s Republic of China (liuch569@mail2.sysu.edu.cn; yedx3@mail.sysu.edu.cn; yuanmd3@mail2.sysu.edu.cn; zhangsm53@mail2.sysu.edu.cn)
James McLaughlin
Affiliation:
Mathematics Department, West Chester University, West Chester, PA, USA (jmclaughlin2@wcupa.edu)
Dongxi Ye
Affiliation:
School of Mathematics (Zhuhai), Sun Yat-Sen University, Zhuhai, Guangdong, People’s Republic of China (liuch569@mail2.sysu.edu.cn; yedx3@mail.sysu.edu.cn; yuanmd3@mail2.sysu.edu.cn; zhangsm53@mail2.sysu.edu.cn)
Miaodan Yuan
Affiliation:
School of Mathematics (Zhuhai), Sun Yat-Sen University, Zhuhai, Guangdong, People’s Republic of China (liuch569@mail2.sysu.edu.cn; yedx3@mail.sysu.edu.cn; yuanmd3@mail2.sysu.edu.cn; zhangsm53@mail2.sysu.edu.cn)
Sumeng Zhang
Affiliation:
School of Mathematics (Zhuhai), Sun Yat-Sen University, Zhuhai, Guangdong, People’s Republic of China (liuch569@mail2.sysu.edu.cn; yedx3@mail.sysu.edu.cn; yuanmd3@mail2.sysu.edu.cn; zhangsm53@mail2.sysu.edu.cn)
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Abstract

This work characterizes the vanishing of the Fourier coefficients of all CM (Complex Multiplication) eta quotients. As consequences, we recover Serre’s characterization about that of $\eta(12z)^{2}$ and recent results of Chang on the pth coefficients of $\eta(4z)^{6}$ and $\eta(6z)^{4}$. Moreover, we generalize the results on the cases of weight 1 to the setting of binary quadratic forms.

Keywords

CM eta quotientsFourier coefficientsvanishingness

MSC classification

Primary: 11F11: Holomorphic modular forms of integral weight 11F20: Dedekind eta function, Dedekind sums 11F30: Fourier coefficients of automorphic forms
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 66 , Issue 4 , November 2023 , pp. 1202 - 1216
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

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