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; it can be expressed in terms of ramification in the $\ell$-adic representation associated to $A$ for any prime $\ell \ne p$.

The conductor of an abelian variety over a number field is defined similarly; it is an ideal that is a product of positive 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 $C$.

Isogenous abelian varieties have the same conductor.

**Authors:**

**Knowl status:**

- Review status: reviewed
- Last edited by Bjorn Poonen on 2022-03-24 16:09:30

**Referred to by:**

- ag.bad_prime
- ag.good_reduction
- g2c.169.a.169.1.bottom
- g2c.25913.a.25913.1.bottom
- g2c.277.a.top
- g2c.2916.b.11664.1.bottom
- g2c.3319.a.3319.1.bottom
- g2c.336.a.172032.1.bottom
- g2c.440509.a.440509.1.bottom
- g2c.invariants
- g2c.label
- g2c.local_invariants
- g2c.paramodular_conjecture
- rcs.rigor.g2c
- rcs.source.g2c
- lmfdb/genus2_curves/main.py (line 981)
- lmfdb/genus2_curves/templates/g2c_browse.html (line 12)
- lmfdb/genus2_curves/templates/g2c_curve.html (line 55)
- lmfdb/genus2_curves/web_g2c.py (line 660)
- lmfdb/modular_curves/main.py (line 328)
- lmfdb/modular_curves/templates/modcurve.html (line 69)

**History:**(expand/hide all)

- 2022-03-24 16:09:30 by Bjorn Poonen (Reviewed)
- 2017-06-30 22:35:08 by Andrew Sutherland (Reviewed)

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