On the U_p-operator in characteristic p

TL;DR

This paper investigates the properties of the U-operator on modular forms over fields of characteristic p, establishing surjectivity under certain conditions and extending recent results to broader settings.

Contribution

It proves the surjectivity of the U-operator on Katz modular forms for specific subgroups and characteristics, generalizing previous findings.

Findings

U-operator induces surjection for all weights k ≥ p+2

Results apply to various subgroups including upper-triangular matrices

Extends known results to broader characteristic and subgroup settings

Abstract

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.