The **conductor** of an abelian variety $A$ over $\Q$ is a positive integer $N$ whose prime factors are the primes $p$ where $A$ has bad reduction. The power to which $p$ divides $N$ depends on the type of bad reduction, and is related to the ramification in the $p$-torsion field of the abelian variety.

The conductor of an abelian variety over a number field is defined similarly; it is an ideal that is a product of non-trivial powers of the prime ideals where $A$ has bad reduction. The conductor of a curve $C$ over a number field is defined to be the conductor of the Jacobian of $C$.

The conductor is an invariant of the isogeny class of an abelian variety.

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- Last edited by Andrew Sutherland on 2017-06-30 22:35:08

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- 2017-06-30 22:35:08 by Andrew Sutherland (Reviewed)