Bounds on Faltings’s delta function through covers

Abstract

Let $X$ be a compact Riemann surface of genus $g_{X}\geq1$. In 1984, G. Faltings introduced a new invariant $\delta_{\operatorname{Fal}}(X)$ associated to $X$. In this paper we give explicit bounds for $\delta_{\operatorname{Fal}}(X)$ in terms of fundamental differential geometric invariants arising from $X$, when $g_{X}>1$. As an application, we are able to give bounds for Faltings’s delta function for the family of modular curves $X_{0}(N)$ 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 $J_{0}(N)$ associated to $X_{0}(N)$.