Component groups of abelian varieties

(Un exposé de Siegfried Bosch au séminaire de théorie des nombres de Jussieu le 6 mai 1999)

Résumé :

Let K be the field of fractions of a discrete valuation ring R and A_K an abelian variety over K. Following Néron and Raynaud, there is the notion of a Néron model for A_K. It is a smooth R-group scheme of finite type A representing all morphisms from generic fibres of smooth R-schemes to A_K. Due to Néron, such a model A exists always, and the group of components of its special fibre is referred to as the component group \Phi _{A} of A_K.

In SGA 7 Grothendieck has studied the problem of tranferring the duality between abelian varieties A_K, A'_K to the level of associated Néron models A, A'. The problem consists in extending the Poincaré bundle on A_K\times _K A'_K to a G_{m,R}-bundle on the Néron model A \times _{R} A'. Surprisingly, the obstruction for doing this is encoded into a bilinear pairing \Phi _{A} \times \Phi _{A'} ---> Q/Z, which Grothendieck conjectured to be perfect.

In the lecture we will discuss Grothendieck's pairing and the cases in which it is known to be perfect. On the other hand, using the technique of Weil restriction in conjunction with results of Edixhoven, we will construct a family of counterexamples, which has been discovered recently.