# Conductor of an abelian variety

In mathematics, in Diophantine geometry, the **conductor of an abelian variety** defined over a local or global field *F* is a measure of how "bad" the bad reduction at some prime is. It is connected to the ramification in the field generated by the division points.

For an Abelian variety *A* defined over a field *F* with ring of integers *R*, consider the Néron model of *A*, which is a 'best possible' model of *A* defined over *R*. This model may be represented as a scheme over

- Spec(
*R*)

(cf. spectrum of a ring) for which the generic fibre constructed by means of the morphism

- Spec(
*F*) → Spec(*R*)

gives back *A*. Let *A*^{0} denote the open subgroup scheme of the Néron model whose fibres are the connected components. For a residue field *k*, *A*^{0}_{k} is a group variety over *k*, hence an extension of an abelian variety by a linear group. This linear group is an extension of a torus by a unipotent group. Let *u* be the dimension of the unipotent group and *t* the dimension of the torus. The order of the conductor is

where δ is a measure of wild ramification.

## Properties

- If
*A*has good reduction then*f*=*u*=*t*= δ = 0. - If
*A*has semistable reduction or, more generally, acquires semistable reduction over a Galois extension of*F*of degree prime to*p*, the residue characteristic, then δ = 0. - If
*p*> 2*d*+ 1, where*d*is the dimension of*A*, then δ = 0.

## References

- S. Lang (1997).
*Survey of Diophantine geometry*. Springer-Verlag, 70-71. ISBN 3-540-61223-8. - J.-P. Serre; J. Tate (1968). "Good reduction of Abelian varieties".
*Ann. Math.***88**: 492-517.