Conductor of An Abelian Variety
Get Conductor of An Abelian Variety essential facts below. View Videos or join the Conductor of An Abelian Variety discussion. Add Conductor of An Abelian Variety to your topic list for future reference or share this resource on social media.
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.


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


(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


  • 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.


  • 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.

  This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.



Music Scenes