On p.56 of the following paper $p$-adic Heights of Heegner points on Shimura curves, the author, Daniel Disegni, claims that $A_s$, which has CM by the order $\mathcal{O}_E[\wp] = \mathcal{O}_F + \wp^s \mathcal{O}_E$, admits $A_s[\mathfrak{p}]$ as the canonical submodule with respect to the reduction $\bmod w$ for a fixed prime $w$ of $H$ (and of $H_s$) above $\wp$.
Now what's intriguing me is that $\mathfrak{p}$ is a prime in $E$, so then perhaps I should interpret $A_s[\mathfrak{p}]$ as $A_s[\mathfrak{p} \cap \mathcal{O}_E[\wp]]$? However, I think that $\mathfrak{p} \cap \mathcal{O}_E[\wp] = \wp O_F + \wp^s O_E$ and that this is "the unique prime above $\wp$" in $\mathcal{O}_E[\wp]$. So this canonical submodule really doesn't depend on the prime $w$?
