Structure theorem for mod $p^m$ singular Siegel modular forms

We prove that all mod $p^m$ singular forms of level $N$, degree $n+r$, and $p$-rank $r$ with $n\ge r$ are congruent mod $p^m$ to linear combinations of theta series of degree $r$ attached to quadratic forms of some level. Moreover, we prove that, the levels of theta series are of the form ``$p-power\times N$’'. Additionally, in some cases of mod $p$ singular forms with smallest possible weight, we prove that the levels of theta series should be $p$.