Congruences involving the U_ℓ operator for weakly holomorphic modular forms

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, … , 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.