A relation between the value of the L-series of the curve: \(y^ 2=x^ 3-k^ 3\) in \(s=1\) and its Selmer group
type: Item
instance of: scholarly article
title: A relation between the value of the L-series of the curve: \(y^ 2=x^ 3-k^ 3\) in \(s=1\) and its Selmer group
zbMATH Open document ID: 0575.14027
DOI: 10.1007/BF01275575
published in: Archiv der Mathematik
publication date: 1985-01-01T00:00:00Z
review text: Let \(E_ k\) be the elliptic curve defined over \({\mathbb{Q}}\) by the equation \(y^ 2=x^ 3-k^ 3,\) where \(k\equiv 1 (mod 4)\), squarefree and prime to 6. The author verifies a weak form of the Birch and Swinnerton-Dyer conjecture on the value of the L-series of the curve \(E_ k\) at \(s=1\) [see \textit{J. Coates} and \textit{A. Wiles}, Invent. Math. 39, 223-251 (1977; Zbl 0359.14009)]. In fact he proves that the k-th Fourier coefficient of a well-known related modular form of weight 3/2 is divisible by 3 if and only if the Tate-Shafarevich group of \(E_ k\) has elements of order 3. By means of Galois cohomology it is shown that the latter is the case if and only if the order of the class group of \({\mathbb{Q}}(\sqrt{-k})\) is divisible by 3.
Mathematics Subject Classification ID: 14H45
Mathematics Subject Classification ID: 14G10
Mathematics Subject Classification ID: 12G05
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 11F11
zbMATH DE Number: 3918549
zbMATH Keywords: evaluation of L-series
zbMATH Keywords: Birch Swinnerton-Dyer conjecture
zbMATH Keywords: elliptic curve
zbMATH Keywords: Galois cohomology
reviewed by: Roelof J. Stroeker
MaRDI profile type: Publication
cites work: On the conjecture of Birch and Swinnerton-Dyer
cites work: The rank of the solutions of \(Y^2=X^3\pm p^3\) over \(\mathbb Q\)
cites work: CYCLOTOMIC FIELDS AND MODULAR CURVES
cites work: Sur les coefficients de Fourier des formes modulaires de poids demi-entier
full work available at URL: https://doi.org/10.1007/bf01275575
OpenAlex ID: W2077986444
author: Gerhard Frey
description: scientific article; zbMATH DE number 3918549