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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 ([email protected])
Chang Liu
Affiliation:
School of Mathematics (Zhuhai), Sun Yat-Sen University, Zhuhai, Guangdong, People’s Republic of China ([email protected]; [email protected]; [email protected]; [email protected])
James McLaughlin
Affiliation:
Mathematics Department, West Chester University, West Chester, PA, USA ([email protected])
Dongxi Ye
Affiliation:
School of Mathematics (Zhuhai), Sun Yat-Sen University, Zhuhai, Guangdong, People’s Republic of China ([email protected]; [email protected]; [email protected]; [email protected])
Miaodan Yuan
Affiliation:
School of Mathematics (Zhuhai), Sun Yat-Sen University, Zhuhai, Guangdong, People’s Republic of China ([email protected]; [email protected]; [email protected]; [email protected])
Sumeng Zhang
Affiliation:
School of Mathematics (Zhuhai), Sun Yat-Sen University, Zhuhai, Guangdong, People’s Republic of China ([email protected]; [email protected]; [email protected]; [email protected])
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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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References

Chan, H. H., Cooper, S. and Liaw, W. -C., On $\unicode{x03B7}^3(a\unicode{x03C4})\unicode{x03B7}^3(b\unicode{x03C4})$ with $a+b=8$, J. Aust. Math. Soc. 84(3) (2008), 301–313.Google Scholar
Chang, S., Complex multiplication of two eta-products, Colloq. Math. 159(1) (2020), 7–24.CrossRefGoogle Scholar
Cox, D., Primes of the Form x 2 + ny 2 –Fermat, Class Field Theory, and Complex multiplication (AMS Chelsea Publishing, Providence, RI, 2022).Google Scholar
Deligne, P. and Serre, J. -P., Formes modulaires de poids 1, Ann. Sci. Ec. Norm. Super. 4 Serie 7 (1974), 507–530.Google Scholar
Huber, T., Mc Laughlin, J. and Ye, D., Lacunary eta quotients with identically vanishing coefficients, Int. J. Number Theory 19(7) (2023), 1639–1670.CrossRefGoogle Scholar
Kani, E., The space of binary theta series, Ann. Sci. Math. Quebec 36(2) (2012), 501–534.Google Scholar
Kani, E., Binary theta series and modular forms with complex multiplication, Int. J. Number Theory 10(4) (2014), 1025–1042.CrossRefGoogle Scholar
Martin, Y., Multiplicative η-quotients, Trans. Amer. Math. Soc. 348(12) (1996), 4825–4856.CrossRefGoogle Scholar
Ribet, K. A., Galois representations attached to eigenforms with Nebentypus Modular Functions of One Variable, V (Proc. Second Internat. Conf. Univ. Bonn, Bonn, 1976). Lecture Notes in Mathematics, Volume 601, (Springer, Berlin, 1977).Google Scholar
Serre, J. -P., Quelques applications du theoreme de densite de Chebotarev, Publ. Math. Inst. Hautes Études Sci.(54)(1981), 323–401.Google Scholar
Serre, J. -P., Sur la lacunarit’e des puissances de η, Glasg. Math. J. 27 (1985), 203–221.CrossRefGoogle Scholar
Wang, X. and Pei, D., Modular Forms With Integral and half-Integral weights. (Science Press, Springer, Beijing, Heidelberg, 2012).CrossRefGoogle Scholar