Effective bounds for Faltings’s delta function

Jay Jorgenson; Jürg Kramer

Annales de la faculté des sciences de Toulouse Mathématiques (2014)

  • Volume: 23, Issue: 3, page 665-698
  • ISSN: 0240-2963

Abstract

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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.

How to cite

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Jorgenson, Jay, and Kramer, Jürg. "Effective bounds for Faltings’s delta function." Annales de la faculté des sciences de Toulouse Mathématiques 23.3 (2014): 665-698. <http://eudml.org/doc/275322>.

@article{Jorgenson2014,
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 $\delta _\{\mathrm\{Fal\}\}(X)$. For a given compact Riemann surface $X$ of genus $g_\{X\}=g$, the invariant $\delta _\{\mathrm\{Fal\}\}(X)$ is roughly given as minus the logarithm of the distance with respect to the Weil-Petersson metric of the point in the moduli space $\{\cal M\}_\{g\}$ of genus $g$ curves determined by $X$ to its boundary $\partial \{\cal M\}_\{g\}$. In this paper we begin by revisiting a formula derived in [14], which gives $\delta _\{\mathrm\{Fal\}\}(X)$ in purely hyperbolic terms provided that $g&gt;1$. This formula then enables us to deduce effective bounds for $\delta _\{\mathrm\{Fal\}\}(X)$ 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.},
author = {Jorgenson, Jay, Kramer, Jürg},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {Faltings's delta function; effective bounds; hyperbolic geometry},
language = {eng},
number = {3},
pages = {665-698},
publisher = {Université Paul Sabatier, Toulouse},
title = {Effective bounds for Faltings’s delta function},
url = {http://eudml.org/doc/275322},
volume = {23},
year = {2014},
}

TY - JOUR
AU - Jorgenson, Jay
AU - Kramer, Jürg
TI - Effective bounds for Faltings’s delta function
JO - Annales de la faculté des sciences de Toulouse Mathématiques
PY - 2014
PB - Université Paul Sabatier, Toulouse
VL - 23
IS - 3
SP - 665
EP - 698
AB - 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 $\delta _{\mathrm{Fal}}(X)$. For a given compact Riemann surface $X$ of genus $g_{X}=g$, the invariant $\delta _{\mathrm{Fal}}(X)$ is roughly given as minus the logarithm of the distance with respect to the Weil-Petersson metric of the point in the moduli space ${\cal M}_{g}$ of genus $g$ curves determined by $X$ to its boundary $\partial {\cal M}_{g}$. In this paper we begin by revisiting a formula derived in [14], which gives $\delta _{\mathrm{Fal}}(X)$ in purely hyperbolic terms provided that $g&gt;1$. This formula then enables us to deduce effective bounds for $\delta _{\mathrm{Fal}}(X)$ 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.
LA - eng
KW - Faltings's delta function; effective bounds; hyperbolic geometry
UR - http://eudml.org/doc/275322
ER -

References

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