Poincaré biextension and idèles on the algebraic curve

The Weil pairing of two element from the torsion of the Jacobian of an algebraic curve may be given by a product of local Hilbert symbols of two special idèles associated to the torsion elements of the Jacobian. On the other hand Arbarello, de Concini and Kac have constructed some central extension of the group of idèles on an algebraic curve, in which the commutator is also equal up to sign to the product of all the local Hilbert symbols of two idèles. The goal of this paper is to explain the reason of this similarity. In fact there is a relation between the Poincaré biextension on the square of Jacobian, defining the Weil pairing, and the central extension constructed by Arbarello, de Concini and Kac. The latter turns out to be the quotient of some biextension of the square of the group of idèles associated to the central extension.