Congruences involving the U operator for weakly holomorphic modular forms

Dohoon Choi, Subong Lim

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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
Publication statusPublished - 2019 Jan 1



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

ASJC Scopus subject areas

  • Algebra and Number Theory

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