Comparison of Arakelov and Poincaré metrics on \(X_0(N)\)
Let \(N\) be a square-free positive integer. This paper proves an estimate comparing the Arakelov metric for the smooth projective curve \(X_0(N)\) with the Poincaré metric \((dx\wedge dy)/y^2\) on its universal covering space, the upper half plane \(\mathcal H\). More specifically, for \(N\) as above let \(g\) be the genus of \(X_0(N)\), let \(f_1,\dots,f_g\) be an orthonormal basis for \(\mathcal S(2,\Gamma_0(N))\) under the Petersson inner product, and let \(F_N=\frac{y^2}g\sum_{i=1}^g| f_i| ^2\). This is the ratio between the metrics of Arakelov and Poincaré. The paper shows that for all \(\varepsilon>0\) there exists \(A_\varepsilon>0\) such that for all \(N\) as above and all \(z\in\mathcal H\), the inequality \(F_N(z)\leq A_\varepsilon N^{1+\varepsilon}\) holds. An intermediate result in the above proof is the following. For all \(\varepsilon>0\) there is a constant \(C_\varepsilon>0\) such that for all \(N\) as above, all newforms \(f\) for \(\Gamma_0(N)\) (required to be normalized eigenforms for the Hecke operators), and all \(z\in\mathcal H\), the inequality \(| yf(z)| \leq C_\varepsilon N^{1/2+\varepsilon}\) holds. Finally, as a corollary of the main theorem, the following two assertions are equivalent: (i) there is an upper bound, polynomial in the conductor, for the degree of a strong modular parametrization of a semi-stable elliptic curve \(E\) over \(\mathbb{Q}\) (strong means that the induced map \(H_1(X_0(N),\mathbb{Z})\to H_1(E,\mathbb{Z})\) is surjective); and (ii) the modular height of such an elliptic curve satisfies an upper bound that is linear in \(\log N\).
modular parametrization of a semi-stable elliptic curve
scientific article; zbMATH DE number 878415
scientific article