Let $K$ be a number field with Galois group $G$ and $N$ be a finitely generated abelian group which is also a discrete $G$-module. Let $D(N)$ be the algebraic group defined as
$D(N)(R)=Hom_{\mathbb{Z}}(N,R^{*})$ for any $k$-algebra $R$.
Then $D(N)$ is an algebraic group of multiplicative type over $K$. Does the Neron model for $D(N)$ exist ? I understand that if $N$ is torsion free as an abelian group, then $D(N)$ is an algebraic torus and such a Neron model exists. But I am not sure about the general case. Thank you very much.
I'm not sure about the state of the general theory, but one approach to this kind of question is to write the group as the kernel of a morphism of tori, to bootstrap this to a morphism of the corresponding Neron models, and then to take the kernel of this morphism.
E.g. in the case of $\mu_n$, which is the kernel of $\mathbb G_m \to \mathbb G_m$ given by raising to the $n$th power, which extends to raising to the $n$th power on Neron models. The kernel is then just $\mu_n$ again. If $n$ is coprime to the residue characteristic, this seems pretty natural. Otherwise (e.g. if $n = p =$ the residue characteristic) then it still seems reasonable, but note that we can no longer really test it by the Neron mapping property (because it is not smooth in this case).