Match module-structure classes with embedding conjugacy classes

Determine which of the isomorphism classes of left \(\mathcal O_p\otimes_{\mathbb Z_p}\mathcal O_p\)-module structures classified in Lemma \ref{lem:mod-str-counting} correspond to the five embeddings \(\varphi_j\) listed in Theorem \ref{thm:5-emb}.

Background

In the ramified case $p\mid\Delta$, the paper classifies superspecial $O_B$-abelian surfaces by reducing the problem to the classification of $\mathrm{GL}_2(\mathcal O_p)$-conjugacy classes of embeddings $$\mathcal O_p\hookrightarrow \mathrm{Mat}_2(\mathcal O_p)$$. Theorem \ref{thm:5-emb} identifies exactly five such conjugacy classes, represented by $\varphi_1,\ldots,\varphi_5$.

A separate bimodule-theoretic analysis identifies the corresponding $\mathcal O_p\otimes_{\mathbb Z_p}\mathcal O_p$-module structures through pairs $(L,C)$, where $L$ is one of three $\mathcal R$-lattices and $C$ belongs to an associated orbit space. Lemma \ref{lem:mod-str-counting} establishes that these orbit spaces contain $3+1+1=5$ isomorphism classes, matching the number of embedding classes, but the precise correspondence between the two classifications is left unresolved in the remark. The paper records the candidate correspondence in equations \eqref{eq:e26}--\eqref{eq:e27} but explicitly omits its details.