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) &rarr; 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, 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


 * $$ f_P = 2u_P + t_P + \delta_P, \, $$

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 &delta;P = 0.
 * If $$p> 2d+1$$, where d is the dimension of A, then $$\delta_P=0$$.
 * If $$p\le 2d+1$$ and F is a finite extension of $$\mathbb{Q}_p$$ of ramification degree $$e(F/\mathbb{Q}_p)$$, there is an upper bound expressed in terms of the function $$L_p(n)$$, which is defined as follows:
 * Write $$n=\sum_{k\ge0}c_kp^k$$ with $$0\le c_k<p$$ and set $$L_p(n)=\sum_{k\ge0}kc_kp^k$$. Then


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


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