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, A⁰_k 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_P be the dimension of the unipotent group and t_P 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

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