The Main Results

In this section, we summarize the main contributions of this paper. First we recall the precise definition of the component group of an abelian variety, then we state our main theorem.

Let $R$ be a discrete valuation ring with field of fractions $K$ and maximal ideal  $\mathfrak{m}$, and let $k=R/\mathfrak{m}$ be the residue class field. Let $A$ be an abelian variety over $K$.

Definition 2.1 (Néron model)   A Néron model of $A$ is a smooth commutative group scheme  $\mathcal{A}$ over $R$ such that $A$ is its generic fiber and  $\mathcal{A}$ satisfies the Néron mapping property: the restriction map

$$\Hom_R(S,\mathcal{A})\longrightarrow \Hom_K(S_K,A)$$

is bijective for all smooth schemes $S$ over $R$.

The Néron mapping property implies that  $\mathcal{A}$ is unique up to a unique isomorphism, so we will refer without hesitation to ``the'' Néron model of $A$. Néron models are separated and of finite type as opposed to just locally of finite type, even though their universal property is on the category of arbitrary smooth $R$-schemes. For more about Néron models see [2].

The closed fiber  $\mathcal{A}_k$ of  $\mathcal{A}$ is a group scheme over $k$, which need not be connected. Denote by  $\mathcal{A}_k^0$ the connected component of $\mathcal{A}_k$ that contains the identity. We have an exact sequence

$$0\longrightarrow \mathcal{A}_k^0\longrightarrow \mathcal{A}_k\longrightarrow \Phi_A\longrightarrow 0,$$

where $\Phi_A$ is a finite étale group scheme over $k$. Equivalently, $\Phi_A$ is a commutative finite group equipped with a continuous action of $\Gal(\overline{k}/k)$.

Definition 2.2 (Component group)   The component group of an abelian variety $A$ over $K$ is the group scheme $\Phi_A=\mathcal{A}_k/\mathcal{A}_k^0$.