Self-intersection of the relative dualizing sheaf on modular curves X 1 ( N )

Hartwig Mayer[1]

  • [1] Universität Regensburg Universitätsstrasse 31 93053 Regensburg, Germany

Journal de Théorie des Nombres de Bordeaux (2014)

  • Volume: 26, Issue: 1, page 111-161
  • ISSN: 1246-7405

Abstract

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Let N be an odd and squarefree positive integer divisible by at least two relative prime integers bigger or equal than 4 . Our main theorem is an asymptotic formula solely in terms of N for the stable arithmetic self-intersection number of the relative dualizing sheaf for modular curves X 1 ( N ) / ℚ . From our main theorem we obtain an asymptotic formula for the stable Faltings height of the Jacobian J 1 ( N ) / ℚ of X 1 ( N ) / ℚ , and, for sufficiently large N , an effective version of Bogomolov’s conjecture for X 1 ( N ) / ℚ .

How to cite

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Mayer, Hartwig. "Self-intersection of the relative dualizing sheaf on modular curves $X_1(N)$." Journal de Théorie des Nombres de Bordeaux 26.1 (2014): 111-161. <http://eudml.org/doc/275788>.

@article{Mayer2014,
abstract = {Let $N$ be an odd and squarefree positive integer divisible by at least two relative prime integers bigger or equal than $4$. Our main theorem is an asymptotic formula solely in terms of $N$ for the stable arithmetic self-intersection number of the relative dualizing sheaf for modular curves $X_1(N)/\mathbb\{Q\}$. From our main theorem we obtain an asymptotic formula for the stable Faltings height of the Jacobian $J_1(N)/\mathbb\{Q\}$ of $X_1(N)/\mathbb\{Q\}$, and, for sufficiently large $N$, an effective version of Bogomolov’s conjecture for $X_1(N)/\mathbb\{Q\}$.},
affiliation = {Universität Regensburg Universitätsstrasse 31 93053 Regensburg, Germany},
author = {Mayer, Hartwig},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Arakelov self-intersection number; relative dualizing sheaf; modular curve},
language = {eng},
month = {4},
number = {1},
pages = {111-161},
publisher = {Société Arithmétique de Bordeaux},
title = {Self-intersection of the relative dualizing sheaf on modular curves $X_1(N)$},
url = {http://eudml.org/doc/275788},
volume = {26},
year = {2014},
}

TY - JOUR
AU - Mayer, Hartwig
TI - Self-intersection of the relative dualizing sheaf on modular curves $X_1(N)$
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2014/4//
PB - Société Arithmétique de Bordeaux
VL - 26
IS - 1
SP - 111
EP - 161
AB - Let $N$ be an odd and squarefree positive integer divisible by at least two relative prime integers bigger or equal than $4$. Our main theorem is an asymptotic formula solely in terms of $N$ for the stable arithmetic self-intersection number of the relative dualizing sheaf for modular curves $X_1(N)/\mathbb{Q}$. From our main theorem we obtain an asymptotic formula for the stable Faltings height of the Jacobian $J_1(N)/\mathbb{Q}$ of $X_1(N)/\mathbb{Q}$, and, for sufficiently large $N$, an effective version of Bogomolov’s conjecture for $X_1(N)/\mathbb{Q}$.
LA - eng
KW - Arakelov self-intersection number; relative dualizing sheaf; modular curve
UR - http://eudml.org/doc/275788
ER -

References

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