  For a perfect field \kappa of characteristic p>0, a positive ingeger N not
divisible by p, and an arbitrary subgroup \Gamma of GL_2(Z/NZ), we prove (with
mild additional hypotheses when p\le 3) that the U-operator on the space
M_k(\Gamma/\kappa) of (Katz) modular forms for \Gamma over \kappa induces a
surjection U:M_{k}(\Gamma/\kappa)\rightarrow M_{k'}(\Gamma/\kappa) for all k\ge
p+2, where k'=(k-k_0)/p + k_0 with 2\le k_0\le p+1 the unique integer congruent
to k modulo p. When \kappa=F_p, p\ge 5, N\neq 2,3, and \Gamma is the subgroup
of upper-triangular or upper-triangular unipotent matrices, this recovers a
recent result of Dewar.
