The Closed Fiber of the Néron Model

In this section we recall some terminology associated with closed fibers of Néron models. Let $K$, $R$, and $k$ be as in Section 2, and let $\Phi_A = \mathcal{A}_k/\mathcal{A}^0_k$ be the group scheme of connected components of the closed fiber  $\mathcal{A}_k$. By Chevalley's structure theorem (see [3], or [4] for a modern account), if $K$ is a perfect extension field of $k$ (e.g., $K = \overline{k}$) then there is a unique short exact sequence

$$0 \rightarrow \mathcal{C} \rightarrow \mathcal{A}_K^0 \rightarrow \mathcal{B} \rightarrow 0$$

with $\mathcal{C}$ a smooth affine algebraic $K$-group and $\mathcal{B}$ an abelian variety. Moreover, there is a unique exact sequence

$$0 \rightarrow \mathcal{T} \rightarrow \mathcal{C} \rightarrow \mathcal{U} \rightarrow 0$$

with $\mathcal{T}$ a torus and $\mathcal{U}$ unipotent.

Using the rigidity of tori, one can show that $\mathcal{T}$ is induced by a unique torus in $\mathcal{A}_k^0$. In particular, the condition that $\mathcal{B} = \mathcal{U} = 0$ is equivalent to the condition that $\mathcal{A}_k^0$ be a torus, and the condition that $\mathcal{U} = 0$ is equivalent to the condition that $\mathcal{A}_k^0$ be the extension of an abelian variety by a torus (i.e., be a semi-abelian variety). These conditions can be checked on a geometric closed fiber.

Definition 4.1   The abelian variety $A$ is said to have purely toric reduction if $\mathcal{A}_k^0$ is torus, and to have semistable reduction if $\mathcal{A}_k^0$ is a semi-abelian variety (i.e., $\mathcal{A}_{\overline{k}}^0$ has vanishing unipotent part).