Let X be a compact Riemann surface of genus gX≥1. In 1984, G. Faltings introduced a new invariant δFal(X) associated to X. In this paper we give explicit bounds for δFal(X) in terms of fundamental differential geometric invariants arising from X, when gX>1. As an application, we are able to give bounds for Faltings’s delta function for the family of modular curves X0(N) in terms of the genus only. In combination with work of A. Abbes, P. Michel and E. Ullmo, this leads to an asymptotic formula for the Faltings height of the Jacobian J0(N) associated to X0(N).