Bounds on Faltings’s delta function through covers

Jay Jorgenson¹ ; Jürg Kramer²

¹ Department of Mathematics<br/>City College of New York<br/>Convent Avenue at 138th Street<br/>New York, NY 10031<br/>United States
² Institut für Mathematik<br/>Humboldt-Universität zu Berlin<br/>Unter den Linden 6<br/>D-10099 Berlin<br/>Germany

Annals of mathematics, Tome 170 (2009) no. 1, pp. 1-43.

Résumé

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 $X_{0}(N)$.

Détail

DOI : 10.4007/annals.2009.170.1