Conductor of An Abelian Variety
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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 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

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 δP = 0.
  • If p > 2d + 1, where d is the dimension of A, then δP = 0.

References

  • S. Lang (1997). Survey of Diophantine geometry. Springer-Verlag. pp. 70-71. ISBN 3-540-61223-8.
  • J.-P. Serre; J. Tate (1968). "Good reduction of Abelian varieties". Ann. Math. The Annals of Mathematics, Vol. 88, No. 3. 88 (3): 492-517. doi:10.2307/1970722. JSTOR 1970722.

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