Néron model, torsion and ramification

Question

Let $B$ a discrete valuation ring, say for simplicity with residue field of characteristic $0$, and $K$ its quotient field. Assume that I have an abelian variety $A$ over $K$ and let $A'$ be its Néron model over $B$. Let $n>1$ an integer. The $n$-torsion points $A[n]$ of $A$ are a finite étale scheme over $K$, i.e. of the form $\textrm{Spec}(L_1\times\cdots\times L_r)$ for finite field extensions $L_i$ of $K$. We let $B_i$ the integral closure of $B$ in $L_i$.

Can we understand the ramification of the ring extensions $B\subset B_i$ in terms of (the $n$-torsion points of) $A'$?

For instance, if $A'$ is an abelian scheme itself, then there is no ramification. Is the converse also true?

More generally, is there a description of the codifferent ideal of the extensions $B\subset B_i$ in terms of $A'$?

Or is there maybe a theory other than Néron models better suited for studying this ramification?

Answer 1

If, for example, $A[n]$ is already contained in $A(K)$, then there will be no ramification, regardless of whether or not $A'$ is an abelian scheme (although that may well force the special fiber to be semi-stable, i.e., totally multiplicative). On the other hand, if you assume that there is no ramification for all $n$, or indeed for all $n=\ell^r$ for $r\ge1$, then $A'$ will be an abelian scheme. This is (more-or-less) the criterion of Néron-Ogg-Shafarevich (as proved by Serre and Tate).