Filtrations of dc-weak eigenforms
The notions of strong, weak and dc-weak eigenforms mod $p^n$ were introduced and studied by Chen, Kiming and Wiese (2013). Here we prove that there can be no uniform weight bound (that is, depending only on $p,n$) on dc-weak eigenforms mod $p^n$ of fixed level when $n \geq 2$. This is in contrast with the result of Kiming, Rustom and Wiese (2016) which establishes a uniform weight bound on strong eigenforms mod $p^n$. As a step towards studying weight bounds for weak eigenforms mod $p^n$, we provide a criterion which allows us to detect whether a given dc-weak eigenform mod $p^n$ is weak.