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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.

Contents

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 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 = 2 u P + t P + δ P ,

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

f = ∏ P P f P .

Properties

  • A has good reduction at P if and only if u P = t P = 0 (which implies f P = δ P = 0 ).
  • A has semistable reduction if and only if u P = 0 (then again δ 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

    Conductor of an abelian variety Wikipedia
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