Poincaré biextension and idèles on an algebraic curve

Abstract

The Weil pairing of two elements of the torsion of the Jacobian of an algebraic curve can be expressed in terms of the product of the local Hilbert symbols of two special idèles associated with the torsion elements of the Jacobian. On the other hand, Arbarello, De Concini, and Kac have constructed a central extension of the group of idèles on an algebraic curve in which the commutator is also equal up to a sign to the product of all the local Hilbert symbols of two idèles. The aim of the paper is to explain this similarity. It turns out that there exists a close connection between the Poincaré biextension over the square of the Jacobian defining the Weil pairing and the central extension constructed by Arbarello, de Concini, and Kac. The latter is a quotient of a certain biextension associated with the central extension.