scholarly journals Zagier duality and integrality of Fourier coefficients for weakly holomorphic modular forms

2015 ◽  
Vol 155 ◽  
pp. 139-162 ◽  
Author(s):  
Yichao Zhang
Keyword(s):  
Modular Forms ◽  
Fourier Coefficients ◽  
Weakly Holomorphic Modular Forms ◽  
Holomorphic Modular Forms

scholarly journals On some automorphic properties of Galois traces of class invariants from generalized Weber functions of level 5

2019 ◽  
Vol 17 (1) ◽  
pp. 1631-1651
Author(s):  
Ick Sun Eum ◽  
Ho Yun Jung
Keyword(s):  
Modular Forms ◽  
Fourier Coefficients ◽  
Singular Values ◽  
Congruence Subgroups ◽  
Maass Form ◽  
Weakly Holomorphic Modular Forms ◽  
Reciprocity Law ◽  
Class Invariants ◽  
Higher Genus ◽  
Holomorphic Modular Forms

Abstract After the significant work of Zagier on the traces of singular moduli, Jeon, Kang and Kim showed that the Galois traces of real-valued class invariants given in terms of the singular values of the classical Weber functions can be identified with the Fourier coefficients of weakly holomorphic modular forms of weight 3/2 on the congruence subgroups of higher genus by using the Bruinier-Funke modular traces. Extending their work, we construct real-valued class invariants by using the singular values of the generalized Weber functions of level 5 and prove that their Galois traces are Fourier coefficients of a harmonic weak Maass form of weight 3/2 by using Shimura’s reciprocity law.

scholarly journals Differential operators on polar harmonic Maass forms and elliptic duality

2019 ◽  
Vol 70 (4) ◽  
pp. 1181-1207
Author(s):  
Kathrin Bringmann ◽  
Paul Jenkins ◽  
Ben Kane
Keyword(s):  
Modular Forms ◽  
Fourier Coefficients ◽  
Differential Operators ◽  
Poincaré Series ◽  
Maass Forms ◽  
Poincare Series ◽  
Weakly Holomorphic Modular Forms ◽  
Integral Weight ◽  
Harmonic Maass Forms ◽  
Holomorphic Modular Forms

Abstract In this paper, we study polar harmonic Maass forms of negative integral weight. Using work of Fay, we construct Poincaré series which span the space of such forms and show that their elliptic coefficients exhibit duality properties which are similar to the properties known for Fourier coefficients of harmonic Maass forms and weakly holomorphic modular forms.

scholarly journals Divisibility properties for weakly holomorphic modular forms with sign vectors

2016 ◽  
Vol 12 (08) ◽  
pp. 2107-2123 ◽  
Author(s):  
Yichao Zhang
Keyword(s):  
Modular Forms ◽  
Fourier Coefficients ◽  
Hecke Operators ◽  
Orthogonal Direction ◽  
Weakly Holomorphic Modular Forms ◽  
Divisibility Properties ◽  
Holomorphic Modular Forms

In this paper, we prove some divisibility results for the Fourier coefficients of reduced modular forms with sign vectors. More precisely, we generalize a divisibility result of Siegel on constant terms when the weight [Formula: see text], which is related to the weight of Borcherds lifts when [Formula: see text]. In particular, we see that such divisibility of the weight of Borcherds lifts only exists for [Formula: see text]. By considering Hecke operators for the spaces of weakly holomorphic modular forms with sign vectors, we obtain divisibility results in an “orthogonal” direction on reduced modular forms.

scholarly journals ZEROS OF WEAKLY HOLOMORPHIC MODULAR FORMS OF LEVEL 4

2014 ◽  
Vol 10 (02) ◽  
pp. 455-470 ◽  
Author(s):  
ANDREW HADDOCK ◽  
PAUL JENKINS
Keyword(s):  
Modular Forms ◽  
Fundamental Domain ◽  
Fourier Coefficients ◽  
Canonical Basis ◽  
Lower Boundary ◽  
Weakly Holomorphic Modular Forms ◽  
Duality Property ◽  
Almost All ◽  
Holomorphic Modular Forms

Let [Formula: see text] be the space of weakly holomorphic modular forms of weight k and level 4 that are holomorphic away from the cusp at ∞. We define a canonical basis for this space and show that for almost all of the basis elements, the majority of their zeros in a fundamental domain for Γ0(4) lie on the lower boundary of the fundamental domain. Additionally, we show that the Fourier coefficients of the basis elements satisfy an interesting duality property.

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