polarization module (awaiting review)

Let $(A, \iota)$ be an abelian variety with RM by $\iota : \Z_F \to \End(A_{\overline{\Q}})$. The polarization module of $A$ is the $\Z_F$-module $$ \mathcal{PM}(A) = \{ \lambda \in \Hom_{\Z_F}(A, A^{\vee}) : \lambda = \lambda^{\vee} \} \subseteq NS(A). $$ The polarization cone is the cone of polarizations $ \mathcal{PM}^+(A) \subseteq \mathcal{PM}(A)$, which equips the polarization module with a notion of positivity as a $\Z_F$-module. We say that a polarization module is isomorphic to a fractional ideal $J$ of $\Z_F$ if there is an orientation preserving $\Z_F$-module isomorphism $\phi : \mathcal{PM}(A) \to J$, i.e. such that $\phi(\mathcal{PM}^+(A)) = J^+$.