Proof. By Lemma 4.1.(1), we may consider the fractional ideals of \(\mathcal{E} = A[\pi]\) up to linear equivalence. By Lemma 4.1.(2), the association \(I \mapsto I * \phi\) is injective when restricted to kernel ideals. Since \(A[\pi]\) is Gorenstein, every \(A[\pi]\)-ideal is a kernel ideal by Proposition 4.5. Hence, the map \(I \mapsto I * \phi\) from linear equivalence classes of \(A[\pi]\)-ideals is injective.
It remains to prove surjectivity, i.e., we need to show that every isogeny \(u\colon \phi \to \psi\) defined over \(k\) arises from some \(A[\pi]\)-ideal \(I\) (up to linear equivalence) in the sense that \(\mathbb{H}(\psi) = I \mathbb{H}(\phi)\), cf. Lemma 3.6. Since \(\mathbb{H}(\phi) = \prod_{\mathfrak{l} \lhd A} H_{\mathfrak{l}}(\phi)\) (and similarly for \(\psi\)), we work locally.
Let \(T_\mathfrak{l}(\phi)\) be the Tate module of \(\phi\), where \(\mathfrak{l}\lhd A\) is a prime different from \(\mathfrak{p}\), and let \(H_{\mathfrak{l}}(\phi)=\mathrm{Hom}(T_\mathfrak{l}(\phi), A_\mathfrak{l})\) be its \(A_\mathfrak{l}\)-dual. Since \(A[\pi]\) is Gorenstein, by [7], \(T_\mathfrak{l}(\phi)\) is free over \(\mathcal{E}_\mathfrak{l}\) of rank \(1\), and so is \(H_{\mathfrak{l}}(\phi)\) (by a property of Gorenstein rings, cf. [7]. Hence, every \(\mathcal{E}_{\mathfrak{l}}\)-submodule of \(H_{\mathfrak{l}}(\phi)\) is represented by an \(\mathcal{E}_{\mathfrak{l}}\)-ideal.

Next, let \(u\colon \phi\to \psi\) be an isogeny defined over \(k\). It induces a natural inclusion \[\label{eq:incH} H_\mathfrak{l}(\psi) \subseteq H_\mathfrak{l}(\phi)\tag{1}\] as sublattices. Moreover, by the Tate isomorphism \[\label{eq:Tate} \mathrm{Hom}_k(\phi,\psi) \otimes A_{\mathfrak{l}} \xrightarrow{\simeq} \mathrm{Hom}_{A_{\mathfrak{l}}[\pi]}(H_{\mathfrak{l}}(\psi), H_{\mathfrak{l}}(\phi)),\tag{2}\] where the set on the right denotes \((\mathcal{E}_{\mathfrak{l}} =) A_{\mathfrak{l}}[\pi]\)-invariant maps, the inclusion of 1 realises \(H_{\mathfrak{l}}(\psi)\) as an \(\mathcal{E}_{\mathfrak{l}}\)-submodule of \(H_{\mathfrak{l}}(\phi)\). This implies that \[\label{eq:locallideal} H_{\mathfrak{l}}(\psi) = I_{\mathfrak{l}}H_{\mathfrak{l}}(\phi) \text{ for some ideal I_{\mathfrak{l}} \lhd \mathcal{E}_{\mathfrak{l}}.}\tag{3}\] Hence, the above gives us such ideals \(I_{\mathfrak{l}}\) for any prime \(\mathfrak{l} \neq \mathfrak{p}\) of \(A\). Moreover, we have \(I_{\mathfrak{l}} = \mathcal{E}_{\mathfrak{l}}\) for all but finitely many \(\mathfrak{l}\), since the isogeny \(u\) has finite kernel \(G\).
Now consider \(\mathfrak{p} \lhd A\) and the Dieudonné module \((H_{\mathfrak{p}}(\phi),f_{\phi,\mathfrak{p}})\); we will drop the \(f_{\phi,\mathfrak{p}}\) from the notation when no confusion can arise. Let \(F_k\) denote the unramified extension of \(F_{\mathfrak{p}}\) with residue field \(k\), and \(\mathcal{O}_k \simeq k[[\mathfrak{p}]]\) its ring of integers, which is a free \(A_{\mathfrak{p}}\)-module of rank \(m = n/d\). Recall that by [5], \(H_{\mathfrak{p}}(\phi)\) is a free (left) \(\mathcal{O}_k\)-module of rank \[\label{eq:OkrankHp} \mathrm{rank}_{\mathcal{O}_k}(H_{\mathfrak{p}}(\phi)) = r.\tag{4}\] Morphisms between Dieudonné modules are a priori \(\mathcal{O}_k\)-module morphisms that commute with \(f_{\phi,\mathfrak{p}}\), which we will call \(\mathcal{O}_k[f_{\phi,\mathfrak{p}}]\)-module homomorphisms.

Let \(u: \phi \to \psi\) be a \(k\)-isogeny as before. By the Tate isomorphism at \(\mathfrak{p}\), (see [5]) we have \[\label{eq:Tateatp} \mathrm{Hom}_k(\phi,\psi) \otimes A_{\mathfrak{p}} \xrightarrow{\simeq} \mathrm{Hom}_{A_{\mathfrak{p}}[\pi]}(H_{\mathfrak{p}}(\psi), H_{\mathfrak{p}}(\phi)).\tag{5}\] Since \(A_{\mathfrak{p}}[\pi] = \mathcal{E}_{\mathfrak{p}}\), Equation 5 shows that the sublattice \(H_{\mathfrak{p}}(\psi)\) is also a (right) \(\mathcal{E}_{\mathfrak{p}}\)-submodule of \(H_{\mathfrak{p}}(\phi)\); thus, \(H_{\mathfrak{p}}(\psi)\) is an \(\mathcal{O}_k[f_{\phi,\mathfrak{p}}]\otimes_{A_{\mathfrak{p}}}\mathcal{E}_{\mathfrak{p}}\)-module.

In particular, taking \(\psi = \phi\), we get that \(H_{\mathfrak{p}}(\phi)\) is itself a (right) \(\mathcal{E}_{\mathfrak{p}}\)-module, which is free since \(A[\pi]\) is Gorenstein, again arguing as in [7]. We now compute its \(\mathcal{E}_{\mathfrak{p}}\)-rank. Since \(\mathcal{O}_k = kA_{\mathfrak{p}}\) is a free \(A_\mathfrak{p}\)-module of rank \(m\), it follows from Equation 4 that \(\mathrm{rank}_{A_{\mathfrak{p}}}(H_{\mathfrak{p}}(\phi)) = mr\). Further, since \(\mathrm{rank}_{A_{\mathfrak{p}}}(\mathcal{E}_{\mathfrak{p}}) = r\), we obtain \[\label{eq:EprankHp} \mathrm{rank}_{\mathcal{E}_{\mathfrak{p}}}(H_{\mathfrak{p}}(\phi)) = m.\tag{6}\] When \(m=1\), or equivalently when \(k=\mathbb{F}_\mathfrak{p}\), we get that \(\mathcal{O}_k \simeq A_{\mathfrak{p}}\), hence \(\mathcal{O}_k[f_{\phi,\mathfrak{p}}] \otimes_{A_{\mathfrak{p}}} \mathcal{E}_{\mathfrak{p}} \simeq \mathcal{E}_{\mathfrak{p}}\), and every \(\mathcal{E}_{\mathfrak{p}}\)-submodule of \(H_{\mathfrak{p}}(\phi)\) is given by an ideal. So in this case, for an isogeny \(u\colon \phi \to \psi\) we get \[\label{eq:localpideal} H_{\mathfrak{p}}(\psi) = I_{\mathfrak{p}} H_{\mathfrak{p}}(\phi).\tag{7}\] as for \(\mathfrak{l}\) above. We then note that the local ideals obtained in 7 and 3 for all \(\mathfrak{l} \neq \mathfrak{p}\) are the localisations of a global lattice, which is closed under the action of \(\mathcal{E}\), and therefore is a global ideal \(I\). This ideal satisfies \(\mathbb{H}(\psi) = I \mathbb{H}(\phi)\) by construction, as required.
Suppose now that \(m > 1\). We do not immediately impose that \(\phi\) is ordinary, but below we will indicate precisely when this assumption is needed to conclude the proof.
Recall that a priori the endomorphism algebra \(D = \widetilde{F} = F(\pi)\) of \(\phi\) satisfies \[\label{eq:Dsplit} D_{\mathfrak{p}} := D \otimes_F F_{\mathfrak{p}} = \bigoplus_{\nu \vert \mathfrak{p}} D_{\nu},\tag{8}\] where \(\nu\) runs over all places of the maximal \(A\)-order \(B\) in \(D\) which lie over \(\mathfrak{p}\). This splitting thus descends to \(B\): \[\label{eq:Bsplit} B_{\mathfrak{p}} = \bigoplus_{\nu \vert \mathfrak{p}} B_{\nu}.\tag{9}\] Here each \(B_{\nu}\) is a local ring with unique maximal ideal \(\mathfrak{p}_{B_{\nu}}\). Among \(\nu \vert \mathfrak{p}\), we distinguish the place \(\widetilde{\mathfrak{p}}\), being the unique place of \(D = \widetilde{F}\) lying over the place \((\pi)\) of \(K = \mathbb{F}_q(\pi)\).

The endomorphism ring \(\mathcal{E} = \mathrm{End}_k(\phi) = A[\pi]\) is an order contained in \(B\), but not necessarily equal to it. Hence, the splitting of 9 may not descend to \(\mathcal{E}\). But since \(\mathcal{E}_{\mathfrak{p}}\) is complete and semilocal, we do obtain the splitting \[\label{eq:Esplit} \mathcal{E}_{\mathfrak{p}} = \bigoplus_{\mathfrak{P} \vert \mathfrak{p}} \mathcal{E}_{\mathfrak{P}},\tag{10}\] where the direct sum runs over the maximal ideals \(\mathfrak{P} = \mathfrak{p}_{B_{\nu}} \cap \mathcal{E}_{\mathfrak{p}}\) of \(\mathcal{E}_{\mathfrak{p}}\); all maximal ideals of \(\mathcal{E}_{\mathfrak{p}}\) arise as such intersections, but those for different \(\nu\) may coincide (cf. [6] or [4]). This splitting in turn induces an alternative splitting \(D_{\mathfrak{p}} = \oplus_{\mathfrak{P} \vert \mathfrak{p}} D_{\mathfrak{P}}\).
Let \(\widetilde{\mathfrak{P}} = \widetilde{\mathfrak{p}} \cap \mathcal{E}_{\mathfrak{p}}\). By Corollary 2.9 (or alternatively by [8]), \(\mathcal{E} = A[\pi]\) is locally maximal at \(\pi\), i.e., \(\mathcal{E}_{\widetilde{\mathfrak{P}}} = B_{\widetilde{\mathfrak{p}}}\) is maximal and \(D_{\widetilde{\mathfrak{P}}} = D_{\widetilde{\mathfrak{p}}}\). We obtain \[\begin{align} \mathcal{E}_{\mathfrak{p}} = \mathcal{E}_{\widetilde{\mathfrak{P}}} \oplus \left( \bigoplus_{\mathfrak{P} \neq \widetilde{\mathfrak{P}}} \mathcal{E}_{\mathfrak{P}} \right) & = B_{\widetilde{\mathfrak{p}}} \oplus \mathcal{E}'_{\mathfrak{p}}, \\ D_{\mathfrak{p}} = D_{\widetilde{\mathfrak{P}}} \oplus \left( \bigoplus_{\mathfrak{P} \neq \widetilde{\mathfrak{P}}} D_{\mathfrak{P}} \right) & = D_{\widetilde{\mathfrak{p}}} \oplus D'_{\mathfrak{p}}. \end{align}\]

For the Dieudonné module \(H_{\mathfrak{p}}(\phi)\), as in Proposition 4.5, cf. [5], we have a decomposition \[H_{\mathfrak{p}}(\phi) = H_{\mathfrak{p}}^c(\phi) \oplus H_{\mathfrak{p}}^{\mathrm{\acute{e}t}}(\phi)\] into its connected component \(H_{\mathfrak{p}}^c(\phi)\) (of positive slope) and its maximal étale quotient \(H_{\mathfrak{p}}^{\mathrm{\acute{e}t}}(\phi)\) (of slope zero). There is a corresponding decomposition of the Frobenius \[f_{\phi,\mathfrak{p}} = f_{\phi,\widetilde{\mathfrak{p}}} \oplus f'_{\phi,\mathfrak{p}} = \bigoplus_{\mathfrak{P} \vert \mathfrak{p}}f_{\phi,\mathfrak{P}}\] such that \[\mathcal{E}_{\widetilde{\mathfrak{P}}} = B_{\widetilde{\mathfrak{p}}} \simeq \mathrm{End}(H_{\mathfrak{p}}^c(\phi), f_{\phi, \widetilde{\mathfrak{p}}}), \qquad \mathcal{E}'_{\mathfrak{p}} \simeq \mathrm{End}(H_{\mathfrak{p}}^{\mathrm{\acute{e}t}}(\phi), f'_{\phi,\mathfrak{p}}).\] Moreover, \(H_{\mathfrak{p}}^c(\phi)\) is a free \(B_{\widetilde{\mathfrak{p}}}\)-module and \(H_{\mathfrak{p}}^{\mathrm{\acute{e}t}}(\phi)\) is a free \(\mathcal{E}'_{\mathfrak{p}}\)-module.
Using the decompositions above, our task is now to describe \((\mathcal{O}_k[f_{\phi,\widetilde{\mathfrak{p}}}] \otimes_{A_{\mathfrak{p}}} B_{\widetilde{\mathfrak{p}}})\)-submodules of \(H^c_{\mathfrak{p}}(\phi)\), resp. \((\mathcal{O}_k[f'_{\phi,\mathfrak{p}}] \otimes_{A_{\mathfrak{p}}} \mathcal{E}'_{\mathfrak{p}})\)-submodules of \(H^{\mathrm{\acute{e}t}}_{\mathfrak{p}}(\phi)\), as ideals, viz. \(I_{\widetilde{\mathfrak{p}}}H^c_{\mathfrak{p}}(\phi)\), resp. \(I'_{\mathfrak{p}} H^{\mathrm{\acute{e}t}}_{\mathfrak{p}}(\phi)\). Using again that \(\mathcal{O}_k = k A_{\mathfrak{p}}\) is a free \(A_{\mathfrak{p}}\)-module of rank \(m\), we see that \[\mathcal{O}_k \otimes_{A_{\mathfrak{p}}} \mathcal{E}_{\mathfrak{P}} = \bigoplus_{i=1}^{g_{\mathfrak{P}}} k \mathcal{E}_{\mathfrak{P}}\] for any \(\mathfrak{P} \vert \mathfrak{p}\), where \(g_{\mathfrak{P}} = \mathrm{gcd}(f(\mathfrak{P}/\mathfrak{p}),m)\) and where \(f(\mathfrak{P}/\mathfrak{p})\) is the residue degree of \(\mathfrak{P}\). Thus, for each \(\mathfrak{P} \vert \mathfrak{p}\), the \((\mathcal{O}_k[f_{\phi,\mathfrak{P}}] \otimes_{A_{\mathfrak{p}}} \mathcal{E}_{\mathfrak{P}})\)-submodules are \(f_{\phi,\mathfrak{P}}\)-stable modules of the form \(I_1 \oplus \ldots \oplus I_{g_{\mathfrak{P}}}\) where each \(I_i\) is a fractional ideal in its respective summand \(k \mathcal{E}_{\mathfrak{P}}\).
To check stability under \(f_{\phi,\mathfrak{p}}\), we must first represent its action on \(H_{\mathfrak{p}}(\phi)\), which we do rationally and locally for each place \(\mathfrak{P} \vert \mathfrak{p}\), closely following [6] and [5]. To represent \(V_\mathfrak{p}(\phi):= H_\mathfrak{p}(\phi)\otimes_{A_\mathfrak{p}}F_\mathfrak{p}\) itself, we consider the decomposition \[F_k \otimes_{F_{\mathfrak{p}}} D_{\mathfrak{p}} \simeq \bigoplus_{\mathfrak{P} \vert \mathfrak{p}} (F_k \otimes_{F_{\mathfrak{p}}} D_{\mathfrak{P}}).\] By construction, this algebra has a left \(F_k\)-action and a right \(D_{\mathfrak{p}}\)-action; recall that \(F_k = \mathrm{Frac}(\mathcal{O}_k)\) is the unramified extension of \(F_{\mathfrak{p}}\) of degree \(m\) with residue field \(k\).
First we consider \(\widetilde{\mathfrak{P}} \vert \mathfrak{p}\). Here the local endomorphism ring \(\mathcal{E}_{\widetilde{\mathfrak{P}}} = B_{\widetilde{\mathfrak{p}}}\) is maximal by the above, so we may argue as in [6]: The field \(F_k\cap D_{\widetilde{\mathfrak{P}}}\) is an unramified extension of \(F_{\mathfrak{p}}\) of degree \(g_{\widetilde{\mathfrak{P}}}\), and the field \(F_k D_{\widetilde{\mathfrak{P}}}\) is an unramified extension of \(D_{\widetilde{\mathfrak{P}}}\) of degree \(\mathrm{lcm}(m, f(\widetilde{\mathfrak{P}}/\mathfrak{p}))/f(\widetilde{\mathfrak{P}}/\mathfrak{p}) = \frac{m \cdot f(\widetilde{\mathfrak{P}}/\mathfrak{p})}{g_{\widetilde{\mathfrak{P}}}\cdot f(\widetilde{\mathfrak{P}}/\mathfrak{p})} = \frac{m}{g_{\widetilde{\mathfrak{P}}}}\). Hence, we see that \(F_k\otimes_{F_\mathfrak{p}} D_{\widetilde{\mathfrak{P}}}\) is a direct sum of \(g_{\widetilde{\mathfrak{P}}}\) copies of the compositum \(F_kD_{\widetilde{\mathfrak{P}}}\): \[F_k \otimes_{F_{\mathfrak{p}}} D_{\widetilde{\mathfrak{P}}} \xrightarrow{\simeq} F_kD_{\widetilde{\mathfrak{P}}} \oplus \ldots \oplus F_kD_{\widetilde{\mathfrak{P}}}.\] Let \(\sigma\) be the Frobenius of \(F_k\) over \(F_{\mathfrak{p}}\). The map giving the above identification is \[\label{eq1} \omega\otimes \beta\longmapsto (\omega\beta, \sigma(\omega)\beta, \dots, \sigma^{g_{\widetilde{\mathfrak{P}}}-1}(\omega)\beta).\tag{11}\] An element \(\lambda \in F_k\) acts on the direct sum by the diagonal matrix \[\mathrm{diag}(\lambda, \sigma(\lambda), \ldots, \sigma^{g_{\widetilde{\mathfrak{P}}}-1}(\lambda)).\] Furthermore, \(\sigma\) acting on the \(F_k\)-factor of the tensor product takes \(\omega\otimes \beta\) to \[\label{eq2} (\sigma(\omega)\beta, \sigma^2(\omega)\beta, \dots, \sigma^{g_{\widetilde{\mathfrak{P}}}}(\omega)\beta).\tag{12}\] Note that \(\sigma\) has order \(m\), hence \(\sigma^{g_{\widetilde{\mathfrak{P}}}}\) has order \(m/g_{\widetilde{\mathfrak{P}}}\), which is the residue degree of \(F_k D_{\widetilde{\mathfrak{P}}}\) over \(D_{\widetilde{\mathfrak{P}}}\). Also note that the Frobenius \(\mathrm{Fr}_{\widetilde{\mathfrak{P}}}\) of \(F_kD_{\widetilde{\mathfrak{P}}}\) over \(D_{\widetilde{\mathfrak{P}}}\) fixes \(D_{\widetilde{\mathfrak{P}}}\). Thus, \(\sigma^{g_{\widetilde{\mathfrak{P}}}}(\omega)\beta = \mathrm{Fr}_{\widetilde{\mathfrak{P}}}(\omega\beta)\), so \(\sigma\) acts by a cyclic permutation followed by \(\mathrm{Fr}_{\widetilde{\mathfrak{P}}}\) in the last place. We now use [9] evaluated on \(\pi = \tau^n\) to obtain the equality \[f(\widetilde{\mathfrak{P}}/\mathfrak{p})\cdot \mathrm{ord}_{\widetilde{\mathfrak{P}}}(\pi) = \frac{n\cdot f(\widetilde{\mathfrak{P}}/\mathfrak{p}) e(\widetilde{\mathfrak{P}}/\mathfrak{p})}{d\cdot H(\phi)} = \frac{n\cdot [D_{\widetilde{\mathfrak{P}}}:F_\mathfrak{p}]}{d\cdot H(\phi)} = m,\] where \(e(\widetilde{\mathfrak{P}}/\mathfrak{p})\) is the ramification index of \(\widetilde{\mathfrak{P}}\) and where the last equality follows from our assumption that \(D\) is commutative. Hence, \(m/g_{\widetilde{\mathfrak{P}}}\) divides \(\mathrm{ord}_{\widetilde{\mathfrak{P}}}(\pi)\). By local class field theory, if \(K_s\) is the unramified extension of a local field \(K\) of degree \(s\), then \[\mathrm{Nr}_{K_s/K}(K_s^\times) = \{a\in K^\times\;\colon\; s\vert\mathrm{ord}_K(a)\}=\mathcal{O}_K^\times\cdot {\varpi_K}^{s\mathbb{Z}}.\] Hence, the element \(\pi\in D_{\widetilde{\mathfrak{P}}}\) is the norm of some element \(\alpha\in F_kD_{\widetilde{\mathfrak{P}}}\): \[\pi=\mathrm{Nr}_{F_kD_{\widetilde{\mathfrak{P}}}/D_{\widetilde{\mathfrak{P}}}}(\alpha) =\alpha\cdot \mathrm{Fr}_{\widetilde{\mathfrak{P}}}(\alpha)\cdots\mathrm{Fr}_{\widetilde{\mathfrak{P}}}^{\frac{m}{g_{\widetilde{\mathfrak{P}}}}-1}(\alpha).\] Let \(u=(1, 1, \dots, \alpha)\in \bigoplus F_kD_{\widetilde{\mathfrak{P}}}\) and define \(f_{\phi, \widetilde{\mathfrak{P}}}=u\sigma\). Then \(f_{\phi, \widetilde{\mathfrak{P}}}\lambda=\lambda^\sigma f_{\phi, \widetilde{\mathfrak{P}}}\) for all \(\lambda\in F_k\) and \[\begin{align} f_{\phi, \widetilde{\mathfrak{P}}}^{m} =(u\sigma)^{m} =u u^{\sigma} \cdots u^{\sigma^{m}-1} \sigma^{m} =u u^{\sigma} \cdots u^{\sigma^{m}-1}. \end{align}\] Now \[\begin{align} & u^\sigma=(1,\dots, \alpha, 1), \dots, u^{\sigma^{{g_{\widetilde{\mathfrak{P}}}}-1}}=(\alpha, 1, \dots, 1), u^{\sigma^{{g_{\widetilde{\mathfrak{P}}}}}}=(1, 1, \dots, \mathrm{Fr}_{\widetilde{\mathfrak{P}}}(\alpha)),\\ & u^{\sigma^{{g_{\widetilde{\mathfrak{P}}}}+1}}=(1,\dots, \mathrm{Fr}_{\widetilde{\mathfrak{P}}}(\alpha), 1), \dots, u^{\sigma^{{2g_{\widetilde{\mathfrak{P}}}}-1}}=(\mathrm{Fr}_{\widetilde{\mathfrak{P}}}(\alpha), 1, \dots, 1), u^{\sigma^{{2g_{\widetilde{\mathfrak{P}}}}}}=(1, 1, \dots, \mathrm{Fr}_{\widetilde{\mathfrak{P}}}^2(\alpha)), \\ & \dots u^{\sigma^{m}-1} = (\mathrm{Fr}_{\widetilde{\mathfrak{P}}}^{\frac{m}{g_{\widetilde{\mathfrak{P}}}}-1}(\alpha), 1, \dots, 1). \end{align}\] We conclude that \[f_{\phi, \widetilde{\mathfrak{P}}}^{m} = u u^{\sigma} \cdots u^{\sigma^{m}-1} =\left(\mathrm{Nr}_{F_kD_{\widetilde{\mathfrak{P}}}/D_{\widetilde{\mathfrak{P}}}}(\alpha), \cdots, \mathrm{Nr}_{F_kD_{\widetilde{\mathfrak{P}}}/D_{\widetilde{\mathfrak{P}}}}(\alpha)\right) = (\pi, \dots, \pi) =\pi.\] Thus, we have constructed the algebra acting on \(\oplus F_kD_{\widetilde{\mathfrak{P}}}\), providing a representation of \(f_{\phi,\widetilde{\mathfrak{P}}}\). As this space has the right dimension, it is isomorphic to the \(\widetilde{\mathfrak{P}}\)-component of \(V_\mathfrak{p}(\phi):= H_\mathfrak{p}(\phi)\otimes_{A_\mathfrak{p}}F_\mathfrak{p}\). On this representation, we may now check \(f_{\phi,\widetilde{\mathfrak{P}}}\)-stability of our fractional ideals.
The above simplifies significantly when we use the assumption that \(\phi\) is ordinary. Indeed, then we have \(H(\phi) = 1\). This implies that \(f(\widetilde{\mathfrak{P}}/\mathfrak{p}) = 1\) since (\(D_{\widetilde{\mathfrak{P}}} = D_{\widetilde{\mathfrak{p}}} =) \widetilde{F}_{\widetilde{\mathfrak{p}}}= F_{\mathfrak{p}}\). So \(g_{\widetilde{\mathfrak{P}}} = 1\) and \(F_k \otimes_{F_{\mathfrak{p}}} D_{\widetilde{\mathfrak{P}}} \simeq F_k D_{\widetilde{\mathfrak{P}}} = F_k\), on which \(f_{\phi, \widetilde{\mathfrak{p}}} = \alpha \sigma\) acts. From \(D_{\widetilde{\mathfrak{P}}} = F_{\mathfrak{p}}\) we also get that its maximal order \(\mathcal{E}_{\widetilde{\mathfrak{P}}}\) equals \(A_{\mathfrak{p}}\), and so \(\mathcal{O}_k\otimes_{A_{\mathfrak{p}}} \mathcal{E}_{\widetilde{\mathfrak{P}}} \simeq \mathcal{O}_k \simeq k[[\mathfrak{p}]]\). The submodules we are considering therefore simplify to \(\mathcal{O}_k\)-fractional ideals \(I_{\widetilde{\mathfrak{P}}}\) which are of the form \(\mathcal{O}_k\langle \mathfrak{p}^\epsilon \rangle\) for some \(\epsilon \geq 0\), i.e. they are generated by \(\mathfrak{p}^{\epsilon}\) as \(\mathcal{O}_k\)-modules. These ideals are visibly stable under \(f_{\phi, \widetilde{\mathfrak{P}}}\). Hence, we find that every \((\mathcal{O}_k[f_{\phi,\widetilde{\mathfrak{P}}}] \otimes_{A_{\mathfrak{p}}} \mathcal{E}_{\widetilde{\mathfrak{P}}})\)-submodule of \(H^c_{\mathfrak{p}}(\phi)\) is given by \(I_{\widetilde{\mathfrak{P}}}H^c_{\mathfrak{p}}(\phi)\), as required.
When \(\phi\) is not ordinary, so \(g_{\widetilde{\mathfrak{P}}} > 1\), there are \({g_{\widetilde{\mathfrak{P}}} \mathrm{ord}_{\widetilde{\mathfrak{P}}}(\pi)/m+1 \choose g_{\widetilde{\mathfrak{P}}}-1}\) inequivalent \((\mathcal{O}_k[f_{\phi,\widetilde{\mathfrak{p}}}] \otimes_{A_{\mathfrak{p}}} \mathcal{E}_{\widetilde{\mathfrak{P}}})\)-submodules of \(H_{\mathfrak{p}}^c(\phi)\).

The arguments of [6] still apply. These show that since \(\mathcal{O}_k \otimes_{A_{\mathfrak{p}}} \mathcal{E}_{\widetilde{\mathfrak{P}}} \simeq \oplus k\mathcal{E}_{\widetilde{\mathfrak{P}}}\) is the maximal order in \(F_k \otimes D_{\widetilde{\mathfrak{P}}} \simeq \oplus F_k \widetilde{F}_{\widetilde{\mathfrak{p}}}\), the \((\mathcal{O}_k \otimes_{A_{\mathfrak{p}}} \mathcal{E}_{\widetilde{\mathfrak{P}}})\)-submodules \(I_1 \oplus \ldots \oplus I_{g_{\widetilde{\mathfrak{P}}}}\) are of the form \((\oplus k\mathcal{E}_{\widetilde{\mathfrak{P}}})\langle \widetilde{\mathfrak{P}}^{\epsilon_1}, \ldots, \widetilde{\mathfrak{P}}^{\epsilon_{g_{\widetilde{\mathfrak{P}}}}} \rangle\). Invariance under \(f_{\phi, \widetilde{\mathfrak{p}}}\) then translates into the condition \[\epsilon_1 \leq \epsilon_2 \leq \ldots \leq \epsilon_{g_{\widetilde{\mathfrak{P}}}} \leq \epsilon_1 + \mathrm{ord}_{\widetilde{\mathfrak{P}}}(\alpha),\] where \(\mathrm{ord}_{\widetilde{\mathfrak{P}}}\) is the extension of the normalised valuation on \(\widetilde{F}_{\widetilde{\mathfrak{p}}}\) to the unramified extension \(F_k \widetilde{F}_{\widetilde{\mathfrak{p}}}\). We note that \[\mathrm{ord}_{\widetilde{\mathfrak{P}}}(\alpha)=\mathrm{ord}_{\widetilde{\mathfrak{P}}}(\pi)/[F_k \widetilde{F}_{\widetilde{\mathfrak{p}}} : \widetilde{F}_{\widetilde{\mathfrak{p}}}]=\frac{\mathrm{ord}_{\widetilde{\mathfrak{P}}}(\pi) \cdot g_{\widetilde{\mathfrak{P}}}}{m}.\] Since \(\mathcal{E}_{\widetilde{\mathfrak{P}}}\) acts diagonally on \(\oplus F_k D_{\widetilde{\mathfrak{P}}}\), after scaling by \(\widetilde{\mathfrak{P}}^{-\epsilon_1}\), we may assume \(\epsilon_1=0\). Hence, the number of \(f_{\phi, \widetilde{\mathfrak{p}}}\)-stable modules up to scaling is the number of integers \[0\leq n_1\leq \cdots\leq n_{g-1}\leq \frac{\mathrm{ord}_{\widetilde{\mathfrak{P}}}(\pi) \cdot g_{\widetilde{\mathfrak{P}}}}{m}.\] This is exactly the binomial coefficient \({g_{\widetilde{\mathfrak{P}}} \mathrm{ord}_{\widetilde{\mathfrak{P}}}(\pi)/m+1 \choose g_{\widetilde{\mathfrak{P}}}-1}\). 0◻

The lemma also explains the above-mentioned example of [3]. Recall that in this case \(\phi\) is supersingular of rank \(2\) over \(k = \mathbb{F}_{q^2}\) defined by \(\phi_T = \alpha \tau^2\), where \(\alpha \in k^{\times}\) is assumed to satisfy \(\alpha\not \in \mathbb{F}_q^\times\) so that \(\widetilde{F}=F(\pi)\) has degree \(2\) over \(F\), and \(\mathfrak{p} = T\). Then \(D = \widetilde{F}\) is commutative and \(\mathcal{E}=k[T]\) is the ring of integers (i.e., the maximal order \(B\)) in \(\widetilde{F}\). Since \(\widetilde{\mathfrak{p}}\) is the only prime above \(\mathfrak{p}\) (cf. [9]), it suffices to determine the \((\mathcal{O}_k[f_{\phi,\widetilde{\mathfrak{p}}}] \otimes_{A_{\mathfrak{p}}} \mathcal{E}_{\widetilde{\mathfrak{p}}})\)-modules in \(H_{\mathfrak{p}}(\phi) = H^c_{\mathfrak{p}}(\phi)\). We have \(f(\widetilde{\mathfrak{p}}/\mathfrak{p})=2\) and \(m=[k:\mathbb{F}_T]=2\), so \(g_{\widetilde{\mathfrak{p}}}=2\). Note that \(T\) remains inert in \(\widetilde{F}\), so \(\mathrm{ord}_{\widetilde{\mathfrak{p}}}(\pi)=\mathrm{ord}_T(T)=1\). By the above, we have \({2 \choose 1}=2\) inequivalent \((\mathcal{O}_k[f_{\phi,\widetilde{\mathfrak{p}}}] \otimes_{A_{\mathfrak{p}}} \mathcal{E}_{\widetilde{\mathfrak{p}}})\)-modules in \(H_{\mathfrak{p}}(\phi)\). In the \(k\)-isogeny class of \(\phi\) we correspondingly have exactly two Drinfeld modules, \(\phi\) and \(\psi\), defined by \(\phi_T\) and \(\psi_T=\alpha^q \tau^2\) respectively, with the isogeny \(\phi\to \psi\) given by \(\tau\). This is in accordance with [3].
Next, we return to the main proof and consider each place \(\mathfrak{P}\vert \mathfrak{p}\) such that \(\mathfrak{P}\neq \widetilde{\mathfrak{P}}\). Assuming \(\phi\) is ordinary, we have \(\mathrm{ord}_\mathfrak{P}(\pi)=0\), so \(\pi\) is a unit in each \(\mathcal{E}_{\mathfrak{P}}\). By a direct adaptation of [6], which applies Hensel’s lemma to the norm form \(\mathrm{Nr}_{(\mathcal{O}_k \otimes \mathcal{E}_{\mathfrak{P})}/\mathcal{E}_{\mathfrak{P}}}\), we find that \(\pi \in \mathcal{E}_{\mathfrak{P}}\) is the norm of some element \(\beta \in \mathcal{O}_k \otimes_{A_{\mathfrak{p}}} \mathcal{E}_{\mathfrak{P}}\), which is thus itself a unit: \[\pi=\mathrm{Nr}_{(\mathcal{O}_k \otimes \mathcal{E}_{\mathfrak{P})}/\mathcal{E}_{\mathfrak{P}}}(\beta).\] As in [6], let \(f_{\phi, \mathfrak{P}}=\beta \sigma\) act on \(F_k \otimes_{F_{\mathfrak{p}}} D_{\mathfrak{P}}\), where as before \(\sigma\) denotes the Frobenius of \(F_k\) over \(F_{\mathfrak{p}}\). As for \(\widetilde{\mathfrak{P}}\), one checks that \(f_{\phi,\mathfrak{P}} \lambda = \lambda^{\sigma} f_{\phi,\mathfrak{P}}\) for all \(\lambda \in F_k\), and that \(f_{\phi,\mathfrak{P}}^m = \pi\). Defining \(v = \oplus_{\mathfrak{P} \neq \widetilde{\mathfrak{P}}} \beta\) and \(f'_{\phi, \mathfrak{p}} = \oplus_{\mathfrak{P} \neq \widetilde{\mathfrak{P}}} f_{\phi, \mathfrak{P}} = v\sigma\) thus provides a suitable representation of the action of Frobenius on \(H_{\mathfrak{p}}^{\mathrm{\acute{e}t}}(\phi) \otimes_{A_{\mathfrak{p}}} F_{\mathfrak{p}}\).
We now argue as in [4] to describe the \((\mathcal{O}_k[f_{\phi,\mathfrak{P}}] \otimes_{A_{\mathfrak{p}}} \mathcal{E}_{\mathfrak{P}})\)-modules for all \(\mathfrak{P} \neq \widetilde{\mathfrak{P}}\). Recall that \(k \simeq \mathbb{F}_{q^n}\) and \(\mathbb{F}_{\mathfrak{p}} \simeq \mathbb{F}_{q^d}\), where \(m = n/d\) is an integer. Consider the Galois group \(G = \mathrm{Gal}(F_k/F_{\mathfrak{p}}) \simeq \mathrm{Gal}(k/\mathbb{F}_{\mathfrak{p}})\), which is cyclic of order \(m\), and has generator \(\sigma\) which on \(k\) acts as \(x \mapsto x^{q^d}\). Then \(G\) acts also on \(\mathcal{O}_k = kA_{\mathfrak{p}} \simeq k[[\mathfrak{p}]]\), and hence on \(\mathcal{O}_k \otimes_{A_{\mathfrak{p}}} \mathcal{E}_{\mathfrak{P}}\), in such a way that \(\mathcal{O}_k^G = A_{\mathfrak{p}}\) and hence \((\mathcal{O}_k \otimes_{A_{\mathfrak{p}}} \mathcal{E}_{\mathfrak{P}})^G = \mathcal{E}_{\mathfrak{P}}\).

We claim that \((\mathcal{O}_k \otimes_{A_{\mathfrak{p}}} \mathcal{E}_{\mathfrak{P}})/\mathcal{E}_{\mathfrak{P}}\) is a \(G\)-Galois extension of rings. By [10], this holds if and only if there exist \(n \in \mathbb{N}\) and \(x_1,\ldots, x_n, y_1, \ldots, y_n \in \mathcal{O}_k \otimes_{A_{\mathfrak{p}}} \mathcal{E}_{\mathfrak{P}}\) such that \[\sum_{i=1}^n x_i \rho(y_i) = \begin{cases} 1 & \text{if } \rho = 1; \\ 0 & \text{ if } \rho \neq 1. \end{cases}\] Since \(G\) acts trivially on \(\mathcal{E}_{\mathfrak{P}}\), it suffices to choose \(x_i, y_i \in \mathcal{O}_k\). Every element \(1 \neq x \in k\) satisfies \[1+x+\ldots+x^{q^n-2} = \sum_{i=1}^{q^n-1} x^{i-1}=0.\] So choose such an \(x \neq 1\), let \(n = q^n-1\), and choose \(x_i = \frac{x^{-i}}{q^n - 1}\) and \(y_i = x^i\) for any \(i = 1, \ldots, q^n-1\). Then \[\sum_{i=1}^{n} x_i y_i = \sum_{i=1}^{q^n-1} \frac{x^{-i}}{q^n - 1} x^i = 1\] and, writing \(\rho= \sigma^j\) for some \(1 \leq j \leq m-1\), \[\sum_{i=1}^n x_i \rho(y_i) = \frac{1}{q^n-1} \sum_{i=1}^{q^n-1} (x^{q^{dj}-1})^i = 0.\] This proves the claim.

Consider the descent datum \(\Phi = \{\Phi_{\rho} = \rho\}_{\rho \in G}\), where we view each \(\rho\) as an \(\mathcal{E}_{\mathfrak{P}}\)-automorphism of \(\mathcal{O}_k \otimes_{A_{\mathfrak{p}}} \mathcal{E}_{\mathfrak{P}}\) by the discussion above. Then by [10], for any \((\mathcal{O}_k \otimes_{A_{\mathfrak{p}}} \mathcal{E}_{\mathfrak{P}})\)-module \(J\) we obtain by descent an \(\mathcal{E}_{\mathfrak{P}}\)-module \[J^{\Phi} = \{ j \in J: \Phi_{\rho}(j) = \rho(j) = j \text{ for all } \rho \in G \} = J^G,\] such that there is a module isomorphism \(J \simeq (\mathcal{O}_k \otimes_{A_{\mathfrak{p}}} \mathcal{E}_{\mathfrak{P}}) \otimes_{\mathcal{E}_{\mathfrak{P}}} J^{\Phi}\).

Now consider a \((\mathcal{O}_k [f_{\phi,\mathfrak{P}}]\otimes_{A_{\mathfrak{p}}} \mathcal{E}_{\mathfrak{P}})\)-module \(J \subseteq F_k \otimes D_{\mathfrak{P}}\). By the above, \(f_{\phi, \mathfrak{P}} = \beta \sigma\) where \(\beta\) is a unit. Hence, \(\beta J = J\) and since \(G = \langle \sigma \rangle\), stability under \(f_{\phi, \mathfrak{P}}\) is equivalent to stability under \(G\). It follows that \(J^G = J \cap D_{\mathfrak{P}}\), where by slight abuse of notation we identify \(D_{\mathfrak{P}}\) with its (diagonal) image in \(F_k \otimes D_{\mathfrak{P}}\). We get that \[J \simeq (\mathcal{O}_k \otimes_{A_{\mathfrak{p}}} \mathcal{E}_{\mathfrak{P}}) \otimes_{\mathcal{E}_{\mathfrak{P}}} J^{G} \simeq \mathcal{O}_k \otimes_{A_{\mathfrak{p}}} \mathcal{J}\] is an extension of the \(\mathcal{E}_{\mathfrak{P}}\)-module \(\mathcal{J} = J \cap D_{\mathfrak{P}}\). And as above, any such \(\mathcal{E}_{\mathfrak{P}}\)-module is an \(\mathcal{E}_{\mathfrak{P}}\)-ideal, which we denote by \(I_{\mathfrak{P}}\).
Collecting all the \(\mathcal{I}_{\mathfrak{P}}\) for \(\mathfrak{P} \neq \widetilde{\mathfrak{P}}\) to obtain \(\mathcal{I}'_{\mathfrak{p}} = \oplus_{\mathfrak{P} \neq \widetilde{\mathfrak{P}}} \mathcal{I}_{\mathfrak{P}}\), we conclude that also every \((\mathcal{O}_k[f'_{\phi,\mathfrak{p}}] \otimes_{A_{\mathfrak{p}}} \mathcal{E}'_{\mathfrak{p}})\)-submodule of \(H^{\mathrm{\acute{e}t}}_{\mathfrak{p}}(\phi)\) is given by \(\mathcal{I}'_{\mathfrak{p}}H^{\mathrm{\acute{e}t}}_{\mathfrak{p}}(\phi)\), as required.

This ends the proof of the theorem. ◻