Abstract
In his seminal paper on arithmetic surfaces Faltings introduced a new invariant associated to compact Riemann surfaces X, nowadays called Faltings’s delta function and here denoted by δ Fal (X). For a given compact Riemann surface X of genus g X =g, the invariant δ Fal (X) is roughly given as minus the logarithm of the distance with respect to the Weil-Petersson metric of the point in the moduli space ℳ g of genus g curves determined by X to its boundary ∂ℳ g . In this paper we begin by revisiting a formula derived in [14], which gives δ Fal (X) in purely hyperbolic terms provided that g>1. This formula then enables us to deduce effective bounds for δ Fal (X) in terms of the smallest non-zero eigenvalue of the hyperbolic Laplacian acting on smooth functions on X as well as the length of the shortest closed geodesic on X. The article ends with a discussion of an application of our results to Parshin’s covering construction.
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