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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 A⁰ 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, 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

fP=2uP+tP+δP,displaystyle f_P=2u_P+t_P+delta _P,,

where δP∈Ndisplaystyle delta _Pin mathbb N is a measure of wild ramification. When F is a number field, the conductor ideal of A is given by

f=∏PPfP.displaystyle f=prod _PP^f_P.

Properties

• 
  A has good reduction at P if and only if uP=tP=0displaystyle u_P=t_P=0 (which implies fP=δP=0displaystyle f_P=delta _P=0).


• 
  A has semistable reduction if and only if uP=0displaystyle u_P=0 (then again δP=0displaystyle 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 p > 2d + 1, where d is the dimension of A, then δ_P = 0.

References

• 
  S. Lang (1997). Survey of Diophantine geometry. Springer-Verlag. pp. 70&ndash, 71. ISBN 3-540-61223-8..mw-parser-output cite.citationfont-style:inherit.mw-parser-output qquotes:"""""""'""'".mw-parser-output code.cs1-codecolor:inherit;background:inherit;border:inherit;padding:inherit.mw-parser-output .cs1-lock-free abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-lock-subscription abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registrationcolor:#555.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration spanborder-bottom:1px dotted;cursor:help.mw-parser-output .cs1-hidden-errordisplay:none;font-size:100%.mw-parser-output .cs1-visible-errorfont-size:100%.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-formatfont-size:95%.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-leftpadding-left:0.2em.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-rightpadding-right:0.2em
  


• 
  J.-P. Serre; J. Tate (1968). "Good reduction of Abelian varieties". Ann. Math. The Annals of Mathematics, Vol. 88, No. 3. 88 (3): 492&ndash, 517. doi:10.2307/1970722. JSTOR 1970722.