Self-intersection of the relative dualizing sheaf on modular curves \(X_{ 1 }(N)\)
The paper under review concerns the Arakelov self-intersection number of the relative dualizing sheaf on modular curves \(X_1(N)\) over \(\mathbb{Q}\). In this paper, the author obtains an asymptotic formula for the self-intersection number with respect to the integer \(N\), from which the author also obtains an asymptotic formula for the stable Faltings height of the Jacobian of \(X_1(N)\) and, for sufficiently large \(N\), an effective version of the Bogomolov's conjecture. Precisely, let \(X_1(N)/\mathbb{Q}\) be the smooth projective algebraic curve that classifies elliptic curves equipped with a point of exact order \(N\). Suppose that \(N\) is odd, squarefree, and of the form \(N=N'qr>0\) with \(q,r\geq 4\) relative prime. Then the minimal regular model \(\mathcal{X}_1(N)/{\mathbb{Z}[\zeta_N]}\) is semistable and the genus \(g_N\) of \(X_1(N)/\mathbb{Q}\) is bigger than \(0\). With the notation \(\overline{\omega}^2_N=\frac{1}{[K:\mathbb{Q}]}\overline{\omega}^2_{\mathcal{X}_1(N)/{\mathcal{O}_K}}\), the stable arithmetic self-intersection number of the dualizing sheaf with the Arakelov metric, the main theorem in the paper under review reads \[ \overline{\omega}^2_N=3g_N\log(N)+o(g_N\log(N)). \] As an application, the stable Faltings height \(h_{\text{Fal}}(J_1(N))\) of the Jacobian of \(X_1(N)\) satisfies the formula \[ h_{\text{Fal}}(J_1(N))=\frac{g_N}{4}\log(N)+o(g_N\log(N)). \] Moreover, for any \(\varepsilon>0\), there is a sufficiently large \(N\) such that the set of algebraic points \[ \{x\in X_1(N)(\overline{\mathbb{Q}}) \mid h_{\text{NT}}(\varphi_D(x))<(\frac{3}{4}-\varepsilon)\log(N)\} \] is finite, and the set of algebraic points \[ \{x\in X_1(N)(\overline{\mathbb{Q}}) \mid h_{\text{NT}}(\varphi_D(x))\leq(\frac{3}{2}+\varepsilon)\log(N)\} \] is finite, if the class \([\mathcal{O}_{X_1(N)}(K_{X_1(N)}-(2g_N-2)D)]\) is a torsion element in \(J_1(N)/\mathbb{Q}\), where \(\varphi_D\) is the embedding \(X_1(N)\to J_1(N)\), \(h_{\text{NT}}\) is the Néron-Tate height and \(K_{X_1(N)}\) is the canonical divisor. This result provides an effective version of the Bogomolov's conjecture for \(X_1(N)\).
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