On the U_p-operator in characteristic p

For a perfect field κof characteristic p>0, a positive ingeger N not divisible by p, and an arbitrary subgroup Γof GL_2(Z/NZ), we prove (with mild additional hypotheses when p\le 3) that the U-operator on the space M_k(Γ/κ) of (Katz) modular forms for Γover κinduces a surjection U:M_{k}(Γ/κ)\rightarrow M_{k'}(Γ/κ) 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 κ=F_p, p\ge 5, N\neq 2,3, and Γis the subgroup of upper-triangular or upper-triangular unipotent matrices, this recovers a recent result of Dewar.

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