The number of elliptic curves over \(\mathbb{Q}\) with conductor \(N\)
type: Item
instance of: scholarly article
title: The number of elliptic curves over \(\mathbb{Q}\) with conductor \(N\)
zbMATH Open document ID: 0868.11029
DOI: 10.1007/BF02567942
author: Armand Brumer
author: Joseph H. Silverman
published in: Manuscripta Mathematica
publication date: 1997-08-19T00:00:00Z
full work available at URL: https://eudml.org/doc/156217
review text: Let \(\varepsilon>0\) and \(S\) be a finite set of rational primes. Put \(M= \prod_{p\in S}p\). The authors prove that there is a constant \(k_1(\varepsilon)\), independent of the set \(S\), such that the number of elliptic curves defined over \(\mathbb{Q}\) with good reduction outside \(S\) is at most \(k_1 (\varepsilon) M^{1/2+\varepsilon}\). In particular, there is a constant \(k_2(\varepsilon)\) such that the number of elliptic curves defined over \(\mathbb{Q}\) with conductor \(N\) is bounded by \(k_2(\varepsilon) N^{1/2+\varepsilon}\). An interesting consequence of this result is that there is a constant \(k_3(\varepsilon)\) such that for all integers \(N\geq 1\), the number of \(\mathbb{Q}\)-rational elliptic factors of the modular Jacobian \(J_0(N)\) is at most \(k_3(\varepsilon) N^{1/2+\varepsilon}\). Furthermore, the authors prove, using standard conjectures, that their bound on the number of elliptic curves defined over \(\mathbb{Q}\) with good reduction outside \(S\) can be improved to \(k_4 M^{k_5/\log\log M}\), where \(k_4\) and \(k_5\) are absolute constants.
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 14H40
zbMATH DE Number: 980701
zbMATH Keywords: number of elliptic curves
zbMATH Keywords: good reduction
zbMATH Keywords: conductor
zbMATH Keywords: elliptic factors of the modular Jacobian
reviewed by: Dimitros Poulakis
MaRDI profile type: Publication
cites work: Q4301629
cites work: The behavior of the Mordell-Weil group of elliptic curves
cites work: Uniform bounds for the number of solutions to Yn = f(X)
cites work: Finiteness theorems for abelian varieties over number fields.
cites work: Q3259107
cites work: Q3735800
cites work: Q4277781
cites work: Q5813236
cites work: An inequality relating the regulator and the discriminant of a number field
cites work: Q3710637
cites work: A quantitative version of Siegel's theorem: integral points on elliptic curves and Catalan curves.
cites work: Algorithm for determining the type of a singular fiber in an elliptic pencil
description: scientific article; zbMATH DE number 980701