2 clarification

I have just realised that a group scheme I've known and loved for years is probably a lot bit wackier than I'd realised.

In this question, in Charles Rezk's answer, I erroneously claim that his construction of the space representing Drinfeld $\Gamma_1(p)$ structures on elliptic curves must be flawed, because the global properties of $Y_1(p)$ that I know from Katz-Mazur seemed to contradict global properties that his construction appeared to me to have. We took the conversation to email and I also started writing down my thoughts more carefully to check there were no problems with them. I found a problem with them---hence this question.

Let $p$ be prime, let $N\geq4$ be an integer prime to $p$, and consider the fine moduli space $Y_1(N)$ over an algebraically closed field $k$ of characteristic $p$. The $N$ isn't important, it just saves me having to use the language of stacks. Let $Y^o$ denote the open affine of $Y_1(N)$ obtained by removing the supersingular points. Over $Y^o$ we have an elliptic curve $E$ (obtained from the universal family over $Y_1(N)$).

Here's

In brief: here's the thing I'm just coming to terms withquestion. Consider The $p$-torsion $E[p]$ of $E$---it's a group scheme and its identity component is non-reduced. But (regarded as an abstract scheme) does it have a component which is reduced? I think it might! This goes against my intuition.