### Abstract

Let λ be an integer, and f(z) = ∑ _{n} _{≫} _{-} _{∞}a(n) q^{n} be a weakly holomorphic modular form of weightλ+12 on Γ (4) with integral coefficients. Let ℓ≥ 5 be a prime. Assume that the constant term a(0) is not zero modulo ℓ. Further, assume that, for some positive integer m, the Fourier expansion of (f|Uℓm)(z)=∑n=0∞b(n)qn has the form (f|Uℓm)(z)≡b(0)+∑i=1t∑n=1∞b(din2)qdin2(modℓ),where d_{1}, … , d_{t} are square-free positive integers, and the operator U_{ℓ} on formal power series is defined by (∑n=0∞a(n)qn)|Uℓ=∑n=0∞a(ℓn)qn.Then, λ≡0(modℓ-12). Moreover, if f~ denotes the coefficient-wise reduction of f modulo ℓ, then we have {limm→∞f~|Uℓ2m,limm→∞f~|Uℓ2m+1}={a(0)θ(z),a(0)θℓ(z)∈Fℓ[[q]]},where θ(z) is the Jacobi theta function defined by θ(z)=∑n∈Zqn2. By using this result, we obtain the distribution of the Fourier coefficients of weakly holomorphic modular forms in congruence classes. This applies to the congruence properties for traces of singular moduli.

Original language | English |
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Pages (from-to) | 671-688 |

Number of pages | 18 |

Journal | Ramanujan Journal |

Volume | 51 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2020 Apr 1 |

### Keywords

- Congruence
- Modular forms of half-integral weight
- Trace of singular moduli

### ASJC Scopus subject areas

- Algebra and Number Theory

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## Cite this

_{ℓ}operator for weakly holomorphic modular forms.

*Ramanujan Journal*,

*51*(3), 671-688. https://doi.org/10.1007/s11139-019-00154-z