Indivisibility of class numbers of imaginary quadratic fields and orders of Tate-Shafarevich groups of elliptic curves with complex multiplication
type: Item
instance of: scholarly article
title: Indivisibility of class numbers of imaginary quadratic fields and orders of Tate-Shafarevich groups of elliptic curves with complex multiplication
zbMATH Open document ID: 0931.11044
author: Ken Ono
author: Winfried Kohnen
published in: Inventiones Mathematicae
publication date: 1999-06-02T00:00:00Z
review text: The authors show that for each odd prime \(l\), each \(\varepsilon>0\) and for all sufficiently large \(X>0\), the number of imaginary quadratic fields of discriminant less than~\(X\) and class number \textit{not} divisible by \(l\) is at least \(\left({{2(l-2)}\over{\sqrt{3}(l-1)}}-\varepsilon\right){{\sqrt{X}}\over {\text{ log} X}}\). They also show that for each elliptic curve \(E\) with complex multiplication, all sufficiently large primes~\(l\) and all sufficiently large \(X>0\), the number of imaginary quadratic fields \(k\) of discriminant less than \(X\) for which the twisted elliptic curve \(E_{(k)}\) has finite Mordell-Weil group \(E_{(k)}({\mathbb Q})\) and has Tate-Shafarevich group of order prime to~\(l\), is at least \(C_{\varepsilon,l}{{\sqrt{X}}\over{\log X}}\) for some constant \(C_{\varepsilon,l}\) that only depends on \(\varepsilon\) and \(l\). The results follow by comparing coefficients of two modular forms of weight \(3/2\) and applying a theorem of J.~Sturm's concerning modular forms modulo~\(p\).
Mathematics Subject Classification ID: 11R11
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11G15
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11F37
zbMATH DE Number: 1275764
zbMATH Keywords: class number
zbMATH Keywords: modular forms of half integral weight
zbMATH Keywords: elliptic curve
zbMATH Keywords: complex multiplication
zbMATH Keywords: imaginary quadratic fields
zbMATH Keywords: finite Mordell-Weil group
zbMATH Keywords: Tate-Shafarevich group
reviewed by: René Schoof
MaRDI profile type: Publication
full work available at URL: s002220050290
OpenAlex ID: W2076093011
DOI: 10.1007/S002220050290
description: scientific article; zbMATH DE number 1275764