Self-intersection of the relative dualizing sheaf on modular curves $X_1(N)$
Hartwig Mayer
TL;DR
The paper develops a comprehensive Arakelov-theoretic analysis of X_1(N) over Q for odd squarefree N with q,r ≥ 4, decomposing the stable arithmetic self-intersection of the relative dualizing sheaf into analytic (via Rankin–Selberg transforms and the Selberg zeta function) and geometric components. It derives an explicit leading-term asymptotic ω_N^2 = 3 g_N log N + o(g_N log N) and translates this into asymptotics for the stable Faltings height h_{Fal}(J_1(N)) = (g_N/4) log N + o(g_N log N), as well as an admissible self-intersection bound and effective Bogomolov-type results for large N. The analysis hinges on precise control of Green's functions at cusps, the hyperbolic/spectral/parabolic Rankin–Selberg contributions, and a meromorphic continuation of a lattice-based zeta function in the appendix. These results advance understanding of arithmetic invariants for modular curves and have potential implications for computing Fourier coefficients of modular forms and explicit height bounds.}
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 $X_1(N) / \mathbb{Q}$.