Suppose $f$ is a mod $p$ modular form of level $N$ with $p>3$ and $p$ not dividing $N$. Is it true that $w(f E_{p+1})=w(f)+p+1$, where $w(f)$ is the filtration of $f$? (The filtration of $f$ is the smallest weight $k$ such that there is a weight $k$ modular form $g$ of level $N$ such that the $q$-expansions of $f$ and $g$ agree $\pmod{p}$.)
I'm guessing is trivial from the geometric perspective, but I just wanted to make sure. (The idea being the zeros of $E_{p+1}$ are disjoint from the zeros of $E_{p-1}$. I just don't really know what it means precisely for $E_{p-1}$ to have a simple zero, or what an order of a zero means in the geometric sense.)
In case anybody else ever asks the same question (though this doesn't seem to be of interest), a more detailed proof of question above is mentioned in http://arxiv.org/abs/1301.3087 (On the theta operator for modular forms modulo prime powers) on page 10, at the middle of the page following the phrase "contradicts the following general fact". So at least it is true.