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

2012/12/06 by Hartwig Mayer, Mayer, Hartwig
Computer Science · Mathematics · #11G18 #11G50 #11M36 #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Number Theory (math.NT) #Polynomial and algebraic computation

paper · doi:10.48550/arxiv.1212.1294

openalex publication_date 2012/12/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04

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.

Citations

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

Related

• On the arithmetic self-intersection number of the dualizing sheaf on arithmetic surfaces
• Faltings modular height and self-intersection of dualizing sheaf
• Self-intersection of the relative dualizing sheaf on modular curves X(N)
• On the arithmetic self-intersection numbers of the dualizing sheaf for Fermat curves of prime exponent
• The intersection matrices of X₀(p^r) and some applications