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If $E$ is an elliptic curve defined over the $p$-adic field $\mathbb Q_p,$ its Tamagawa number is the finite index \[c_p=[E(\mathbb Q_p):E^0(\mathbb Q_p)].\] Here $E^0(\mathbb Q_p)$ is the subgroup of points which have good reduction. If $E$ has good reduction, then $E(\mathbb Q_p)=E^0(\mathbb Q_p)$ and $c_p=1$.

An elliptic curve $E$ defined over $\mathbb Q$ has a Tamagawa number $c_p$ at every prime $p$, with $c_p=1$ for all primes of good reduction (and hence $c_p=1$ for all but finitely many primes).

Knowl status:
  • Review status: reviewed
  • Last edited by John Jones on 2018-06-19 15:32:39
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