On the $U_p$ operator in characteristic $p$

For a perfect field $\kappa$ of characteristic $p>0$, a positive integer $N$ not divisible by $p$, and an arbitrary subgroup $\Gamma$ of $\mathrm{GL}_2(\mathbf{Z} / N \mathbf{Z})$, we prove (with mild additional hypotheses when $p\le 3$) that the $U$-operator on the space $M_k(\mathscr{P}_{\Gamma} / \kappa)$ of (Katz) modular forms for $\Gamma$ over $\kappa$ induces a surjection $U:M_{k}(\mathscr{P}_{\Gamma} / \kappa)\twoheadrightarrow M_{k'}(\mathscr{P}_{\Gamma} / \kappa)$ 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 $\kappa=\mathbf{F}_p$, $p\ge 5$, $N\neq 2,3$, and $\Gamma$ is the subgroup of upper-triangular or upper-triangular unipotent matrices, this recovers a recent result of Dewar [3].

Keywords

Mod $p$ modular forms, Atkin $U_p$-operator

2010 Mathematics Subject Classification

11F33, 11G18

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Published 1 August 2014