On the U_p operator in characteristic p

For a perfect field κ of characteristic p > 0, a positive integer N not divisible by p, and an arbitrary subgroup F of GL2(Z/NZ), we prove (with mild additional hypotheses when p ≤ 3) that the U-operator on the space Mk(PF/κ) of (Katz) modular forms for F over κ induces a surjection U : Mk(PF/κ) →Mk' (PF/κ) for all k ≥ p + 2, where k' = (k - k0)/p + k0 with 2 ≤ k0 ≤ p + 1 the unique integer congruent to k modulo p. When κ = Fp, p ≥ 5, N ≠= 2, 3, and F is the subgroup of upper-triangular or upper-triangular unipotent matrices, this recovers a recent result of Dewar [3].