Quotient by the canonical subgroup of an elliptic curve

Question

This question bothers me! Consider a $p$-adically complete and separated ring $R$, and an elliptic curve $E$ over $R$. It is clear that, if $E$ has ordinary reduction, then there exists a subgroup scheme of $E[p]$, the $p$-torsion of $E$, lifting the kernel of Frobenius. I also know, e.g. see Katz's "p-adic properties of modular schemes and modular forms", that this lifting exists also for elliptic curves not too supersingular. Usually this subgroup is called the canonical subgroup. Now, I see in a lot of papers that it is possible to consider the quotient $E/H$, where this $H$ is the canonical subgroup, but I don't see why this quotient is again an elliptic curve. First, why is the quotient representable by a scheme? And second, why does it satisfies the required properties for being an elliptic curve? Thank you very much for any suggestion.