Let $p$ be a prime. Let $K/\mathbb{Q}\_p$ be a finite extension and $\mathcal{O}=\mathcal{O}\_K$ its ring of integers.


Let $A\_K$ be an abelian variety and $\mathcal{A}\_\mathcal{O}$ denote its Neron model over $\mathcal{O}$. We don't assume that $\mathcal{A}\_\mathcal{O}$ has good or semistable reduction.


Let $\ell$ be a prime different from $p$ and let $G=A[\ell]\_\mathcal{O}$ be the scheme-theoretic kernel of multiplication by $\ell$ in $\mathcal{A}\_\mathcal{O}$. We know that $G\times\_{\text{Spec} ~\mathcal{O}} \text{Spec} ~K = A\_K[\ell]$ is a finite subgroup scheme of order $\ell^{2g}$, where $g$ is the dimension of $A\_K$. 


I have a very stupid question here:


**Is $A[\ell]\_{\mathcal{O}}$ finite and flat? (or at least quasi-finite)? 
If not, are there some necessary conditions for $A[\ell]\_{\mathcal{O}}$ to be finite and flat?**



Finite subgroup scheme and Neron model of an abelian variety
In general the scheme $\mathcal A[l]\_{\mathcal O}$ is not finite because of the following lemma.



> 
> Let $f:X\to Y$ be a separated quasi-finite flat morphism of noetherian schemes. Then it is finite iff the fibral rank is locally constant.
> 
> 
> 


For a proof see the paper "Les schémas de modules des courbes elliptiques" by Deligne and Rappoport (II$.1.19$). 


So, you actually need just to compute the fibral rank of $\mathcal A[l]\_{\mathcal O}$. For example, if $A\_K$ is an elliptic curve with additive or multiplicative reduction it is usually not finite. Note that there is a small mistake in Jason Starr's comment, the rank of $\mathcal A[l]\_k$ might be bigger than $l$ in the case of multiplicative reduction b/c $\mathcal A[l]\_k$ isn't necessary connected.


However, $\mathcal A[l]\_{\mathcal O}$ is always quasi-finite provided that $l$ is invertible on $\mathcal O\_K$. Moreover, it is always étale and you don't need any semi-stability assumptions for this claim. For a proof look at Lemma 7.3/2 in the book "Néron Models" by Bosch, Lutkebohmert, Raynaud.


