Tamagawa Numbers

Let $A$ be an abelian variety over a local field $K$ with residue class field $k$, and let $\mathcal{A}$ be the Néron model of $A$ over the ring of integers of $K$. The closed fiber $\mathcal{A}_{k}$ of $\mathcal{A}$ need not be connected. Let $\mathcal{A}^0_k$ denote the geometric component of $\mathcal{A}$ that contains the identity. The group $\Phi_{\mathcal{A}} = \mathcal{A}_k / \mathcal{A}^0_k$ of connected components is a finite group scheme over $k$. This group scheme is called the component group of $\mathcal{A}$, and the Tamagawa number of $A$ is $c_A = \char93 \Phi_{\mathcal{A}}(k)$.

Now suppose that $A$ is an abelian variety over a global field $K$. For every place $v$ of $K$, the Tamagawa number of $A$ at $v$, denoted $c_{A,v}$ or just $c_v$, is the Tamagawa number of $A_{K_v}$, where $K_v$ is the completion of $K$ at $v$.