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Effective bounds for Faltings’s delta function
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 23 (2014) no. 3, pp. 665-698.

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 (·). 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 g des courbes de genre g déterminé par X à son bord 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.

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 g of genus g curves determined by X to its boundary 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.

@article{AFST_2014_6_23_3_665_0,
     author = {Jay Jorgenson and J\"urg Kramer},
     title = {Effective bounds for Faltings's delta function},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {665--698},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 23},
     number = {3},
     year = {2014},
     doi = {10.5802/afst.1420},
     mrnumber = {3266709},
     zbl = {1327.14127},
     language = {en},
     url = {afst.centre-mersenne.org/item/AFST_2014_6_23_3_665_0/}
}
Jay Jorgenson; Jürg Kramer. Effective bounds for Faltings’s delta function. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 23 (2014) no. 3, pp. 665-698. doi : 10.5802/afst.1420. https://afst.centre-mersenne.org/item/AFST_2014_6_23_3_665_0/

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