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

${\displaystyle f_{P}=2u_{P}+t_{P}+\delta _{P},\,}$

where ${\displaystyle \delta _{P}\in \mathbb {N} }$ is a measure of wild ramification. When F is a number field, the conductor ideal of A is given by

${\displaystyle f=\prod _{P}P^{f_{P}}.}$

Properties

• A has good reduction at P if and only if ${\displaystyle u_{P}=t_{P}=0}$ (which implies ${\displaystyle f_{P}=\delta _{P}=0}$).
• A has semistable reduction if and only if ${\displaystyle u_{P}=0}$ (then again ${\displaystyle \delta _{P}=0}$).
• 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 ${\displaystyle p>2d+1}$, where d is the dimension of A, then ${\displaystyle \delta _{P}=0}$.
• If ${\displaystyle p\leq 2d+1}$ and F is a finite extension of ${\displaystyle \mathbb {Q} _{p}}$ of ramification degree ${\displaystyle e(F/\mathbb {Q} _{p})}$, there is an upper bound expressed in terms of the function ${\displaystyle L_{p}(n)}$, which is defined as follows:

Write ${\displaystyle n=\sum _{k\geq 0}c_{k}p^{k}}$ with ${\displaystyle 0\leq c_{k}<p}$ and set ${\displaystyle L_{p}(n)=\sum _{k\geq 0}kc_{k}p^{k}}$. Then^[1]

${\displaystyle (*)\qquad f_{P}\leq 2d+e(F/\mathbb {Q} _{p})\left(p\left\lfloor {\frac {2d}{p-1}}\right\rfloor +(p-1)L_{p}\left(\left\lfloor {\frac {2d}{p-1}}\right\rfloor \right)\right).}$

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

References

1. Brumer, Armand; Kramer, Kenneth (1994). "The conductor of an abelian variety". Compositio Math. 92 (2): 227-248.

• 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. 88 (3). The Annals of Mathematics, Vol. 88, No. 3: 492–517. doi:10.2307/1970722. JSTOR 1970722.