A Pairing on the Group of Degree Zero Divisors of a Curve Over a Number Field: Pairing Construction on Degree Zero Divisors of a Curve over a Number Field with Arakelov Geometry

Abstract

In this paper we construct a pairingon the group of degree zero divisors of a curve over a number field. This isaccomplished by passing from divisors of the curve to divisors of an associatedscheme and then employing an Arakelov intersection theory. Arakelov geometrystudies a scheme X over the ring of integers Z, by putting Hermitian metrics onholomorphic vector bundles over X(C), the complex points of X. This extraHermitian structure is applied as a substitute, for the failure of the schemeSpec(Z) to be a complete variety..