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) &rarr; 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, A0k is a group variety over k, hence an extension of an abelian variety by a linear group. This linear group is 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 &delta; is a measure of wild ramification.

Properties

 * If A has good reduction then f = u = t = &delta; = 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 &delta; = 0.
 * If p &gt; 2d + 1, where d is the dimension of A, then &delta; = 0.