Conductor of an abelian variety

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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.

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 A with 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 an extension of a torus by a unipotent group. Let uP be the dimension of the unipotent group and tP the dimension of the torus. The order of the conductor at P is

 f_P = 2u_P + t_P + \delta_P , \,

where \delta_P\in\mathbb N is a measure of wild ramification. When F is a number field, the conductor ideal of A is given by

 f= \prod_P P^{f_P}.

Properties[edit]

  • A has good reduction at P if and only if u_P=t_P=0 (which implies f_P=\delta_P= 0).
  • A has semistable reduction if and only if u_P=0 (then again \delta_P= 0).
  • If A acquires semistable reduction over a Galois extension of F of degree prime to p, the residue characteristic at P, then δP = 0.
  • If p > 2d + 1, where d is the dimension of A, then δP = 0.

References[edit]