### 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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Journal | Ramanujan Journal |

DOIs | |

Publication status | Published - 2019 Jan 1 |

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### Keywords

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

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

**Congruences involving the U _{ℓ} operator for weakly holomorphic modular forms.** / Choi, Dohoon; Lim, Subong.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - Congruences involving the Uℓ operator for weakly holomorphic modular forms

AU - Choi, Dohoon

AU - Lim, Subong

PY - 2019/1/1

Y1 - 2019/1/1

N2 - Let λ be an integer, and f(z) = ∑ n ≫ - ∞a(n) qn 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 d1, … , dt are square-free positive integers, and the operator Uℓ 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.

AB - Let λ be an integer, and f(z) = ∑ n ≫ - ∞a(n) qn 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 d1, … , dt are square-free positive integers, and the operator Uℓ 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.

KW - Congruence

KW - Modular forms of half-integral weight

KW - Trace of singular moduli

UR - http://www.scopus.com/inward/record.url?scp=85069820621&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85069820621&partnerID=8YFLogxK

U2 - 10.1007/s11139-019-00154-z

DO - 10.1007/s11139-019-00154-z

M3 - Article

AN - SCOPUS:85069820621

JO - Ramanujan Journal

JF - Ramanujan Journal

SN - 1382-4090

ER -