Variation of the Faltings height in a \(\overline{\mathbb{Q}}\)-isogeny class of elliptic curves
type: Item
instance of: scholarly article
title: Variation of the Faltings height in a \(\overline{\mathbb{Q}}\)-isogeny class of elliptic curves
zbMATH Open document ID: 0952.11018
DOI: 10.1215/S0012-7094-99-09703-X
author: Lucien Szpiro
author: Emmanuel Ullmo
published in: Duke Mathematical Journal
publication date: 2000-03-27T00:00:00Z
review text: This paper proves some bounds on the variation of the Faltings height of a non-CM elliptic curve within a \(\overline{\mathbb Q}\)-isogeny class. The principal result in this direction is the following. Let \(E\) be a semi-stable elliptic curve over a number field \(K\) without complex multiplication. Let \(n\) be a positive integer, and let \(\prod_{i=1}^r p_i^{\alpha_i}\) be its prime factorization. Let \(P\) be a torsion point of \(E\) of order exactly \(n\), and let \(\pi\:E\to E'\) be the \(K(P)\)-isogeny whose kernel is generated by \(P\). Then \[ h(E')=h(E)+\tfrac 12\log n-\sum_i ((p_i^{\alpha_i}-1)/((p_i^2-1)p_i^{\alpha_i-1}))\log p_i + O(1), \] where \(O(1)\) depends only on \(E\). Moreover, if \(\text{Gal}(\overline K/K)\) acts transitively on the set of points of order \(n\) of \(E\), then the \(O(1)\) can be omitted. The paper also proves a similar result for the norm of the conductor.
Mathematics Subject Classification ID: 11G50
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11R32
Mathematics Subject Classification ID: 14G40
zbMATH DE Number: 1425197
zbMATH Keywords: Faltings height
zbMATH Keywords: isogeny class
zbMATH Keywords: Arakelov theory
zbMATH Keywords: conductor
reviewed by: Paul Vojta
MaRDI profile type: Publication
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full work available at URL: s0012-7094-99-09703-x
OpenAlex ID: W2046740849
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description: scientific article; zbMATH DE number 1425197