Non-trivial zeroes for quadratic twists of Hasse-Weil \(L\)-functions
type: Item
instance of: scholarly article
title: Non-trivial zeroes for quadratic twists of Hasse-Weil \(L\)-functions
zbMATH Open document ID: 0999.11029
author: Jorge Jiménez Urroz
published in: Journal of Number Theory
publication date: 2000-09-27T00:00:00Z
review text: Let \(E\) be an elliptic curve defined over \({\mathbb Q}\), endowed with a model \(y^2=x^3+Ax+B\) with \(A,B\in{\mathbb Q}\). This article deals with the problem of the distribution of numbers \(D\) such that the rank of the Mordell-Weil group of the \(D\)-th quadratic twist \(E_D\) is positive. Here sieve techniques lead to lower bounds for the number of fundamental discriminants with prescribed number of prime divisors and bounded absolute value, so that the Hasse-Weil \(L\)-function of \(E_D\) vanishes at \(s=1\).
reviewed by: Federico Pellarin
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11G40
zbMATH DE Number: 1375114
zbMATH Keywords: rank of elliptic curves over \({\mathbb Q}\)
zbMATH Keywords: quadratic twists
zbMATH Keywords: Hasse-Weil \(L\)-functions
zbMATH Keywords: Mordell-Weil group
zbMATH Keywords: sieve techniques
zbMATH Keywords: lower bounds
zbMATH Keywords: number of fundamental discriminants
MaRDI profile type: Publication
full work available at URL: jnth.1999.2391
OpenAlex ID: W2031608652
cites work: The Square-Free Sieve and the Rank of Elliptic Curves
cites work: Q4052222
cites work: Q3856819
cites work: Nonvanishing of \(L\)-series and the combinatorial sieve. With an appendix by David E. Rohrlich: Unboundedness of the Tate-Shafarevich group in families of quadratic twists
cites work: Q4011129
cites work: Q4345954
cites work: Non-vanishing of quadratic twists of modular \(L\)-functions
cites work: Averages of twisted elliptic L-functions
cites work: Q4001553
cites work: On Ranks of Twists of Elliptic Curves and Power-Free Values of Binary Forms
DOI: 10.1006/JNTH.1999.2391
description: scientific article; zbMATH DE number 1375114