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 genus g, for line bundles of degree g equivalence is shown to the height on the Jacobian defined by Theta. We recover the classical formula due to Faltings and Hriljac for the Neron-Tate height on the Jacobian in terms of the intersection pairing on the arithmetic surface.