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**.

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- Last edited by Andrew Sutherland on 2020-10-24 16:30:43

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- 2020-10-24 16:30:43 by Andrew Sutherland (Reviewed)
- 2020-01-03 22:19:49 by Andrew Sutherland (Reviewed)
- 2018-05-24 17:00:16 by John Cremona (Reviewed)

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