Effective bounds for Faltings’s delta function

Jay Jorgenson; Jürg Kramer

doi:10.5802/afst.1420

Jay Jorgenson; Jürg Kramer

Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 23 (2014) no. 3, pp. 665-698.

Abstract
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 $\partial ℳ_{g}$ . 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.

Dans son article fondateur sur les surfaces arithmétiques Faltings a introduit un nouvel invariant des surfaces de Riemann compactes, que l’on appelle de nos jours l’invariant delta de Faltings et que l’on note $δ_{Fal} (\cdot)$ . Pour une surface de Riemann compacte $X$ de genre $g_{X} = g$ , l’invariant $δ_{Fal} (X)$ est donné à peu de choses près par l’opposé du logarithme de la distance, pour la métrique de Weil-Petersson, du point sur l’espace de modules $ℳ_{g}$ des courbes de genre $g$ déterminé par $X$ à son bord $\partial ℳ_{g}$ . Dans le présent article nous commençons par un nouvel examen de la formule obtenue dans [14], qui décrit $δ_{Fal} (X)$ en termes purement hyperboliques, tout au moins si $g > 1$ . Cette formule nous permet ensuite de déduire des bornes effectives pour $δ_{Fal} (X)$ en termes de la plus petite valeur propre non-nulle du Laplacien hyperbolique agissant sur les fonctions lisses sur $X$ et du minimum des longueurs des géodésiques fermées sur $X$ . L’article se termine par une discussion d’une application de nos résultats à la construction du recouvrement de Parshin.

EuDML MR Zbl

DOI: 10.5802/afst.1420

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     author = {Jay Jorgenson and J\"urg Kramer},
     title = {Effective bounds for {Faltings{\textquoteright}s} delta function},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {665--698},
     publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
     address = {Toulouse},
     volume = {Ser. 6, 23},
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     doi = {10.5802/afst.1420},
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Jay Jorgenson; Jürg Kramer. Effective bounds for Faltings’s delta function. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 23 (2014) no. 3, pp. 665-698. doi : 10.5802/afst.1420. https://afst.centre-mersenne.org/articles/10.5802/afst.1420/

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