Résume | Let K be the field of fractions of a discrete valuation ring R and A_Kan abelian variety over K. Following Néron and Raynaud, there isthe notion of a Néron model for A_K. It is a smooth R-group schemeof finite type A representing all morphisms from generic fibres of smoothR-schemes to A_K. Due to Néron, such a model A exists always, andthe group of components of its special fibre is referred to as the componentgroup \Phi _{A} of A_K.In SGA 7 Grothendieck has studied the problem of tranferring the dualitybetween abelian varieties A_K, A'_K to the level of associated Néronmodels A, A'. The problem consists in extending the Poincaré bundleon 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 abilinear pairing \Phi _{A} \times \Phi _{A'} ---> Q/Z, which Grothendieckconjectured to be perfect.In the lecture we will discuss Grothendieck's pairing and the casesin which it is known to be perfect. On the other hand, using the techniqueof Weil restriction in conjunction with results of Edixhoven, we will constructa family of counterexamples, which has been discovered recently. |