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>