Line bundles on arithmetic surfaces and intersection theory

For line bundles on arithmetic varieties we construct height functions using arithmetic intersection theory. In the case of an arithmetic surface, generically of genusg, for line bundles of degreeg equivalence is shown to the height on the Jacobian defined by Θ. We recover the classical formula due to Faltings and Hriljac for the Néron-Tate height on the Jacobian in terms of the intersection pairing on the arithmetic surface.