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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''0 denote the open subgroup scheme of the Néron model whose fibres are the connected components. For a maximal ideal ''P'' of ''A'' with residue field ''k'', ''A''0''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 ''uP'' be the dimension of the unipotent group and ''tP'' the dimension of the torus. The order of the conductor at ''P'' is : where is a measure of wild ramification. When ''F'' is a number field, the conductor ideal of ''A'' is given by : 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Conductor of an abelian variety」の詳細全文を読む スポンサード リンク
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