The theta cycles for modular forms modulo prime powers

Abstract

Abstract 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} .