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 torsion points.

Definition

For an abelian variety A defined over a field F as above, 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 maximal ideal P of R with 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_P be the dimension of the unipotent group and t_P the dimension of the torus. The order of the conductor at P is

where is a measure of wild ramification. When F is a number field, the conductor ideal of A is given by

Properties

• A has good reduction at P if and only if (which implies ).
• A has semistable reduction if and only if (then again ).
• If A acquires semistable reduction over a Galois extension of F of degree prime to p, the residue characteristic at P, then δ_P = 0.
• If , where d is the dimension of A, then .
• If and F is a finite extension of of ramification degree , there is an upper bound expressed in terms of the function , which is defined as follows:

Write with and set . Then

Further, for every with there is a field with and an abelian variety of dimension so that is an equality.

References