Self-intersection of the relative dualizing sheaf on modular curves $X_1(N)$

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 $X_1(N)/\mathbb{Q}$. 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 $X_1(N)/\mathbb{Q}$.

References in corpus (1)

• On the arithmetic self-intersection numbers of the dualizing sheaf for Fermat curves of prime exponent

Cited by in corpus (6)

• Arakelov Self-intersection numbers of minimal regular models of modular curves $X_0(p^2)$
• Semi-stable models of modular Curves $X_0(p^2)$ and some arithmetic applications
• Lower bounds on the arithmetic self-intersection number of the relative dualizing sheaf on arithmetic surfaces
• The minimal regular model of a Fermat curve of odd squarefree exponent and its dualizing sheaf
• An effective Arakelov-theoretic version of the hyperbolic isogeny theorem
• Bounds for canonical Green's function at cusps