en>Cydebot: Robot - Moving category Recursion theory to Computability theory per CFD at Wikipedia:Categories for discussion/Log/2011 February 5.

2011-02-14T20:00:27Z

Robot - Moving category Recursion theory to Computability theory per CFD at Wikipedia:Categories for discussion/Log/2011 February 5.

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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#Abelian_varieties|torsion points]].

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 (mathematics)|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

:<math> f_P = 2u_P + t_P + \delta_P , \, </math>

where <math>\delta_P\in\mathbb N</math> is a measure of wild ramification. When ''F'' is a number field, the conductor ideal of ''A'' is given by
:<math> f= \prod_P P^{f_P}.</math>

==Properties==
* ''A'' has [[good reduction]] at ''P'' if and only if <math>u_P=t_P=0</math> (which implies <math>f_P=\delta_P= 0</math>).
* ''A'' has [[semistable abelian variety|semistable reduction]] if and only if <math>u_P=0</math> (then again <math>\delta_P= 0</math>).
* 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, where ''d'' is the dimension of ''A'', then δP = 0.

==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 journal | author=J.-P. Serre | coauthors=J. Tate | title=Good reduction of Abelian varieties | journal=Ann. Math. | volume=88 | year=1968 | pages=492–517 | doi=10.2307/1970722 | issue=3 | publisher=The Annals of Mathematics, Vol. 88, No. 3 | jstor=1970722 }}

[[Category:Abelian varieties]]
[[Category:Diophantine geometry]]

[[Category:Algebraic number theory]]

{{numtheory-stub}}

Index set (recursion theory) - Revision history

en>Cydebot: Robot - Moving category Recursion theory to Computability theory per CFD at Wikipedia:Categories for discussion/Log/2011 February 5.