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 R 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

[math]\displaystyle{ f_P = 2u_P + t_P + \delta_P , \, }[/math]

where [math]\displaystyle{ \delta_P\in\mathbb N }[/math] is a measure of wild ramification. When F is a number field, the conductor ideal of A is given by

[math]\displaystyle{ f= \prod_P P^{f_P}. }[/math]

Properties

• A has good reduction at P if and only if [math]\displaystyle{ u_P=t_P=0 }[/math] (which implies [math]\displaystyle{ f_P=\delta_P= 0 }[/math]).
• A has semistable reduction if and only if [math]\displaystyle{ u_P=0 }[/math] (then again [math]\displaystyle{ \delta_P= 0 }[/math]).
• 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 [math]\displaystyle{ p\gt 2d+1 }[/math], where d is the dimension of A, then [math]\displaystyle{ \delta_P=0 }[/math].
• If [math]\displaystyle{ p\le 2d+1 }[/math] and F is a finite extension of [math]\displaystyle{ \mathbb{Q}_p }[/math] of ramification degree [math]\displaystyle{ e(F/\mathbb{Q}_p) }[/math], there is an upper bound expressed in terms of the function [math]\displaystyle{ L_p(n) }[/math], which is defined as follows:

Write [math]\displaystyle{ n=\sum_{k\ge0}c_kp^k }[/math] with [math]\displaystyle{ 0\le c_k\lt p }[/math] and set [math]\displaystyle{ L_p(n)=\sum_{k\ge0}kc_kp^k }[/math]. Then

[math]\displaystyle{ (*)\qquad f_P \le 2d + e(F/\mathbb{Q}_p) \left( p \left\lfloor \frac{2d}{p-1} \right\rfloor + (p-1)L_p\left( \left\lfloor \frac{2d}{p-1} \right\rfloor \right) \right). }[/math]

Further, for every [math]\displaystyle{ d,p,e }[/math] with [math]\displaystyle{ p\le 2d+1 }[/math] there is a field [math]\displaystyle{ F/\mathbb{Q}_p }[/math] with [math]\displaystyle{ e(F/\mathbb{Q}_p)=e }[/math] and an abelian variety [math]\displaystyle{ A/F }[/math] of dimension [math]\displaystyle{ d }[/math] so that [math]\displaystyle{ (*) }[/math] is an equality.

References

• S. Lang (1997). Survey of Diophantine geometry. Springer-Verlag. pp. 70–71. ISBN 3-540-61223-8. https://archive.org/details/surveydiophantin00lang_347. 
• 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.