The theta cycles for modular forms modulo prime powers

Recently, Chen and Kiming studied the theta operator on modular forms modulo prime powers p m {p{m}}, where p ≥ 5 {p\geq 5} and m ≥ 2 {m\geq 2}. In this paper, we study mod p m {p{m}} filtrations and mod p m {p{m}} theta cycles. We give a bound on some elements in the mod p m {p{m}} theta cycle (m ≥ 2 {m\geq 2}), and we exactly compute those values in the case that m = 2 {m=2}.