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Conductor of an abelian variety

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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 A 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, Ak 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 = 2 u P + t P + δ P , {\displaystyle f_{P}=2u_{P}+t_{P}+\delta _{P},\,}

where δ P ∈ N {\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

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

Properties

  • A has good reduction at P if and only if u P = t P = 0 {\displaystyle u_{P}=t_{P}=0} (which implies f P = δ P = 0 {\displaystyle f_{P}=\delta _{P}=0} ).
  • A has semistable reduction if and only if u P = 0 {\displaystyle u_{P}=0} (then again δ P = 0 {\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 δP = 0.
  • If p > 2 d + 1 {\displaystyle p>2d+1} , where d is the dimension of A, then δ P = 0 {\displaystyle \delta _{P}=0} .
  • If p ≤ 2 d + 1 {\displaystyle p\leq 2d+1} and F is a finite extension of Q p {\displaystyle \mathbb {Q} _{p}} of ramification degree e ( F / Q p ) {\displaystyle e(F/\mathbb {Q} _{p})} , there is an upper bound expressed in terms of the function L p ( n ) {\displaystyle L_{p}(n)} , which is defined as follows:
Write n = ∑ k ≥ 0 c k p k {\displaystyle n=\sum _{k\geq 0}c_{k}p^{k}} with 0 ≤ c k < p {\displaystyle 0\leq c_{k}<p} and set L p ( n ) = ∑ k ≥ 0 k c k p k {\displaystyle L_{p}(n)=\sum _{k\geq 0}kc_{k}p^{k}} . Then
( ∗ ) f P ≤ 2 d + e ( F / Q p ) ( p ⌊ 2 d p − 1 ⌋ + ( p − 1 ) L p ( ⌊ 2 d p − 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 d , p , e {\displaystyle d,p,e} with p ≤ 2 d + 1 {\displaystyle p\leq 2d+1} there is a field F / Q p {\displaystyle F/\mathbb {Q} _{p}} with e ( F / Q p ) = e {\displaystyle e(F/\mathbb {Q} _{p})=e} and an abelian variety A / F {\displaystyle A/F} of dimension d {\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.
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