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 A⁰ denote the open subgroup scheme of the Néron model whose fibres are the connected components. For a residue field k, A⁰_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

$f=2u+t+\delta ,\,$

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 > 2d + 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.