CONGRUENCES RELATED TO MODULAR FORMS

Abstract

Let f be a modular form of weight k for a congruence subgroup Γ ⊂ SL₂(Z), and t a weight 0 modular function for Γ. Assume that near t = 0, we can write f = ∑_n≥0b_n tⁿ, b_n ∈ Z. Let ℓ(z) be a weight k + 2 modular form with q-expansion ∑γ_nqⁿ, such that the Mellin transform of ℓ can be expressed as an Euler product. Then we show that if for some integers a_i, d_i, then the congruence relation b_mp^r -γ_pb_mp^r-1 + ε_pp^k+1b_mp^r-2 ≡ 0 (mod p^r) holds. We give a number of examples of this phenomena.