Effective bounds for Faltings's delta function

Abstract

We obtain bounds for the Faltings's delta function for any Riemann surface of genus greater than one. The bounds are in terms of the genus of the surface and two basic quantities coming from hyperbolic geometry: The length of the shortest closed geodesic, and the smallest non-zero eigenvalue of the Laplacian which acts on smooth functions. In the case when the surface in question is a finite degree cover of a fixed base surface, then bounds are given in terms of the degree of the curve and data associated to the base surface.