Conductor of an abelian variety explained

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 R 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

f _{P} = 2 u _{P} + t _{P} + \delta _{P} ,

where

\delta _{P\inN} is a measure of wild ramification. When F is a number field, the conductor ideal of A is given by f = \prod _{P} P ^{f _{P}} .

Properties

• A has good reduction at P if and only if

u _{P=t} _{P=0} (which implies f _{P=\delta} _{P=} 0 ).

• A has semistable reduction if and only if

u _{P=0} (then again \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> 2 d +1 , where d is the dimension of A, then \delta _{P=0} .

• If

p\le 2 d +1 and F is a finite extension of Q _{p} of ramification degree e(F/ Q _{p)} , there is an upper bound expressed in terms of the function L _{p(n)} , which is defined as follows:

Write

n = \sum _{k\ge0} c ^{k} _{kp} with 0 \le c _{k<p} and set L _{p(n)=\sum} _{k\ge0} kc ^{k} _{kp} . Then^[1] (*) f _{P} \le 2 d + e(F/ Q _{p)} \left( p \left\lfloor (2 d)/(p -1) \right\rfloor + (p -1 )L _{p\left(} \left\lfloor (2 d)/(p -1) \right\rfloor \right) \right).

Further, for every

d,p,e with p\le 2 d +1 there is a field F/ Q _{p} with e(F/ Q _{p)=e} and an abelian variety A/F of dimension d so that (*) is an equality.

References

• Book: Serge Lang

. S. Lang . Serge Lang . Survey of Diophantine geometry . limited . . 1997 . 3-540-61223-8 . 70 - 71 .

• J.-P. Serre . J. Tate . Good reduction of Abelian varieties . Ann. Math. . 88 . 1968 . 492 - 517 . 10.2307/1970722 . 3 . The Annals of Mathematics, Vol. 88, No. 3 . 1970722 .

Notes and References

1. Brumer . Armand . Kramer . Kenneth . The conductor of an abelian variety . Compositio Math. . 1994 . 92 . 2 . 227-248.