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.