Congruences involving the U operator for weakly holomorphic modular forms

Dohoon Choi, Subong Lim

Research output: Contribution to journalArticle

Abstract

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 U 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 languageEnglish
JournalRamanujan Journal
DOIs
Publication statusPublished - 2019 Jan 1

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Modular Forms
Congruence
Integer
Modulo
Operator
Constant term
Square free
Fourier Expansion
Formal Power Series
Theta Functions
Coefficient
Fourier coefficients
Jacobi
Modulus
Trace
Denote
Zero

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 operator for weakly holomorphic modular forms. / Choi, Dohoon; Lim, Subong.

In: Ramanujan Journal, 01.01.2019.

Research output: Contribution to journalArticle

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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.

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