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

$f_P=2u_P+t_P+{\delta }_P,$

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 ${\delta }_P=0$.
• 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 $n=\sum _{k\ge 0}{c}_k{p}^k$ with $0\le c_k<p$ and set $L_p(n)=\sum _{k\ge 0}k{c}_k{p}^k$. Then

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

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.

References

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