If $R$ is a PID and $M$ a finitely generated torsion module, hence isomorphic to $\bigoplus_{i=1}^n R/f_i$, then the characteristic ideal of $M$ is generated by $\prod_{i=1}^n f_i$. In this case, I believe it agrees with the Fitting ideal. The terminology comes from the case where $V$ is a finite-dimensional vector space over a field $k$, equipped with a $k$-linear endomorphism $A$. Repackaging $V$ as a finitely generated torsion module over the PID $R=k[X]$ (with $X$ acting via $A$), the the characteristic ideal is simply the one generated by the characteristic polynomial of $A$.

@Kevin This onion is perhaps also germane: theonion.com/articles/…

(Presumably, you weren't writing Hailong Dao's answer.)

A warning is worth giving if you're going to use the ``$z^s$ for $s \in \mathcal{O}_{\mathbf{C}_p}$'' description. If $s \notin \mathbf{Z}_p$ then $\chi(z)=z^s$ takes values in $\mathcal{O}_{\mathbf{C}_p}^\times$, not necessarily in $\mathbf{Z}_p^\times$, so you might consider this larger class of characters. And for these more general characters $\chi$, then $\chi$ admits an "exponential" description ($\chi(z)=z^s$) if and only if $|\chi(1+p)-1|_p \leq p^{-1/(p-1)}$. This is related to the $p$-adic radius of convergence of the exponential power series $\exp(X)$ in the last comment.

Moreover, the heuristic that there is something weird about the "theory of the automorphism groups" of inseparable extensions. Rather, the automorphisms that do exist are perfectly fine; it's just that inseparable extensions are more rigid, so there are fewer of them.

@Brian: to be clear, they pass over which issue in silence? The claim that $X^\textrm{an}$ is complex-analytically isomorphic to $\mathfrak{h}^*/\Gamma$?

I'll give this answer the credit because of the explicit citation, but really both this and Scott's were helpful. The better approach than what I wrote above is to take the entire S.E.S. 0 \to Z \to G_m \to E' \to 0 and twist it by K/Q_p (avoiding the extension+restriction of scalars) to get 0 \to Z' \to G_m' \to E \to 0. (Twisting of Z,G_m is done exactly as it is for E.) Taking Galois cohomology and computing, one gets` 0 \to G_m'(Q_p) \to E(Q_p) \to Z/2 \to 0, and if one wants to know E(Q_p)` one is left with explicitly describing G_m'(Q_p), e.g. via the kernel of the norm

Thanks for the reference. I'd love to see Silverman's statement presented as a presentation of analytic groups, though. Any thoughts on that? (BTW, re: the possible index 2, see Scott Carnahan's comment about Galois cohomology.)

Re: your doubts, I'll lump those under "modulo descent".