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Feb 22, 2018 at 22:35 comment added Eric Peterson I don't know about an indirect construction, nor about the dependence on $\pi$, but I believe "admits an $\mathcal O_K$-module structure" is an indirect characterization: there is a unique such formal $\mathcal O_K$-module deformation, rather in the same way that there is a unique deformation of $\widehat{\mathbb G}_m$ as a formal group from $\mathbb F_p$ to $\mathbb Z_p$ (as, in the $p$-adic setting, formal groups are a/k/a formal $\mathcal O_{\mathbb Q_p}$--modules).
Feb 22, 2018 at 2:04 history asked skd CC BY-SA 3.0