Tamagawa numbers (reviewed)

For an abelian variety $A$ over a $p$-adic field, the Tamagawa number of $A$ is the number of connected components of its Néron model.

For a smooth projective curve $X/\Q$ the Tamagawa number of $X$ at at a prime $p$ is the Tamagawa number of the base change of its Jacobian to the field $\Q_p$.

It is a positive integer that is equal to 1 at all primes of good reduction for the Jacobian; it may also be 1 at primes of bad reduction.

The product of the Tamagawa numbers over all primes is a positive integer known as the Tamagawa product.