Conductor of an abelian variety: Difference between revisions
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In [[mathematics]], in [[Diophantine geometry]], the '''conductor of an abelian variety''' defined over a [[local field|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 [[division point]]s. | In [[mathematics]], in [[Diophantine geometry]], the '''conductor of an abelian variety''' defined over a [[local field|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 [[division point]]s. | ||
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:Spec(''F'') → Spec(''R'') | :Spec(''F'') → Spec(''R'') | ||
gives back ''A''. Let ''A''<sup>0</sup> denote the open subgroup scheme of the Néron model whose fibres are the connected components. For a [[residue field]] ''k'', ''A''<sup>0</sup><sub>''k''</sub> is a group variety over ''k'', hence an extension of an abelian variety by a linear group. This linear group | gives back ''A''. Let ''A''<sup>0</sup> denote the open subgroup scheme of the Néron model whose fibres are the connected components. For a [[residue field]] ''k'', ''A''<sup>0</sup><sub>''k''</sub> 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'' be the dimension of the unipotent group and ''t'' the dimension of the torus. The order of the conductor is | ||
:<math> f = 2u + t + \delta , \, </math> | :<math> f = 2u + t + \delta , \, </math> | ||
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* If ''A'' has [[semistable abelian variety|semistable reduction]] or, more generally, acquires semistable reduction over a Galois extension of ''F'' of degree prime to ''p'', the residue characteristic, then δ = 0. | * If ''A'' has [[semistable abelian variety|semistable reduction]] or, more generally, acquires semistable reduction over a Galois extension of ''F'' of degree prime to ''p'', the residue characteristic, then δ = 0. | ||
* If ''p'' > 2''d'' + 1, where ''d'' is the dimension of ''A'', then δ = 0. | * If ''p'' > 2''d'' + 1, where ''d'' is the dimension of ''A'', then δ = 0. | ||
==References== | ==References== | ||
* {{cite book | author=S. Lang | authorlink=Serge Lang | title=Survey of Diophantine geometry | publisher=[[Springer-Verlag]] | year=1997 | isbn=3-540-61223-8 | pages=70-71 }} | *{{cite book | author=S. Lang | authorlink=Serge Lang | title=Survey of Diophantine geometry | publisher=[[Springer-Verlag]] | year=1997 | isbn=3-540-61223-8 | pages=70-71 }} | ||
* {{cite journal | author=J.-P. Serre | coauthors=J. Tate | title=Good reduction of Abelian varieties | journal=Ann. Math. | volume=88 | year=1968 | pages=492-517 }} | *{{cite journal | author=J.-P. Serre | coauthors=J. Tate | title=Good reduction of Abelian varieties | journal=Ann. Math. | volume=88 | year=1968 | pages=492-517 }}[[Category:Suggestion Bot Tag]] | ||
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Latest revision as of 06:00, 1 August 2024
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 division points.
For an Abelian variety A defined over a field F 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 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 u be the dimension of the unipotent group and t the dimension of the torus. The order of the conductor is
where δ is a measure of wild ramification.
Properties
- If A has good reduction then f = u = t = δ = 0.
- If A has semistable reduction or, more generally, acquires semistable reduction over a Galois extension of F of degree prime to p, the residue characteristic, then δ = 0.
- If p > 2d + 1, where d is the dimension of A, then δ = 0.
References
- S. Lang (1997). Survey of Diophantine geometry. Springer-Verlag, 70-71. ISBN 3-540-61223-8.
- J.-P. Serre; J. Tate (1968). "Good reduction of Abelian varieties". Ann. Math. 88: 492-517.