Comparison of Arakelov and Poincaré metrics on \(X_0(N)\)
type: Item
instance of: scholarly article
title: Comparison of Arakelov and Poincaré metrics on \(X_0(N)\)
zbMATH Open document ID: 0895.14007
DOI: 10.1215/S0012-7094-95-08012-0
author: Ahmed Abbes
author: Emmanuel Ullmo
published in: Duke Mathematical Journal
publication date: 1996-06-23T00:00:00Z
review text: 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 \(\log N\).
Mathematics Subject Classification ID: 14G40
Mathematics Subject Classification ID: 11G18
Mathematics Subject Classification ID: 11F25
Mathematics Subject Classification ID: 14G35
Mathematics Subject Classification ID: 14H25
zbMATH DE Number: 878415
zbMATH Keywords: modular height
zbMATH Keywords: Arakelov metric
zbMATH Keywords: Poincaré metric
zbMATH Keywords: modular parametrization of a semi-stable elliptic curve
zbMATH Keywords: modular curve
reviewed by: Paul Vojta
MaRDI profile type: Publication
cites work: Q3125872
cites work: INTERSECTION THEORY OF DIVISORS ON AN ARITHMETIC SURFACE
cites work: Hecke operators on \(\Gamma_0(m)\)
cites work: Sur les représentations $l$-adiques associées aux formes modulaires de Hilbert
cites work: Q3213184
cites work: Q4402450
cites work: Q5190274
cites work: On the Holomorphy of Certain Dirichlet Series
cites work: Q3739268
cites work: Q3972269
cites work: Q4272975
cites work: Modular Parametrizations of Elliptic Curves
cites work: Q3943036
description: scientific article; zbMATH DE number 878415