Self-intersection of the relative dualizing sheaf on modular curves X1(N)

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 X1(N)/Q. From our main theorem we obtain an asymptotic formula for the stable Faltings height of the Jacobian J1(N)/Q of X1(N)/Q, and, for sufficiently large N, an effective version of Bogomolov's conjecture for X1(N)/Q. © Société Arithmétique de Bordeaux, 2014, tous droits réservés.