Conductor of an abelian variety

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) → 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 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 δ is a measure of wild ramification.

Properties

  • If A has good reduction then f = u = t = δ = 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 δ = 0.
  • If p > 2d + 1, where d is the dimension of A, then δ = 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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