### Special fibers of Neron models and wild ramification, by Qing Liu and Dino Lorenzini

This paper has now appeared in Crelle, vol 532 (2001),
and so the preprint has been removed.
Let $K$ be a field with a discrete valuation. Let
${\cal O}_K$ denote the ring of integers of $K$. Let $k$ be the
residue field of ${\cal O}_K$, assumed to be algebraically closed of
characteristic $p \ge 0$.
Let $G/K$ be a commutative group scheme with N\'eron model
${\cal G}/{\cal O}_K$. Let ${\cal G}_k/k$ be the
special fiber of ${\cal G}/{\cal O}_K$, and let ${\cal G}^0_k/k$ denote the
connected component of 0 in ${\cal G}_k$. The
group of components of ${\cal G}$ is the
abelian group $\Phi(G) := {\cal G}_k/{\cal G}^0_k$.
We say that {\it $G/K$ has split reduction} if the extension
$$ 0 \longrightarrow {\cal G}^0_k(k) \longrightarrow
{\cal G}_k(k) \stackrel{c}{\longrightarrow} \Phi(G)
\longrightarrow 0 $$ is split.
The core of this article is a detailed study of the splitting properties
of elliptic curves and of norm tori and their
duals, with applications to abelian varieties
with potentially purely multiplicative reduction.
In all cases studied, we find that there exists a constant $c$ depending
only on the dimension of $G$ such that, if
$G$ has totally not split reduction, then the Swan conductor of $G/K$
is positive and bounded by $c$.
We also find that there is a constant $d$, depending
only on the dimension of $G$, such that $G_M/M$ has split reduction for any
tame extension $M/K$ of degree greater than $d$.

Qing Liu and Dino Lorenzini <liu@math.u-bordeaux.fr, lorenzini@math.uga.edu>