On the rational points of \(y^ 2 = x^ 3 - k\)
type: Item
instance of: scholarly article
title: On the rational points of \(y^ 2 = x^ 3 - k\)
zbMATH Open document ID: 0773.11037
author: Shaowei Zhang
published in: Chinese Science Bulletin
publication date: 1993-05-24T00:00:00Z
review text: The author considers elliptic curves with \(\mathbb{Z}\left[{-1+\sqrt 3\over 2}\right]\) as its ring of complex multiplications and proves some interesting theorems in support of the Birch and Swinnerton-Dyer conjectures regarding elliptic curves with complex multiplication. More specifically he proves the following three theorems: (i) For primes \(p\equiv 2\), \(5\pmod 9\), \(L_ p(1)\) and \(L_{p^ 2}(1)\) are both nonzero, where \(L_ D(s)\) denotes the Hecke \(L\)-series of the curve \(\Gamma_ D:x^ 3+y^ 3=D\) (which, by a simple change of variables takes the more familiar form \(y^ 2=x^ 3-432 D^ 2)\). (ii) For primes \(p\equiv 2\pmod 9\), \(\text{rank}_ \mathbb{Q}(\Gamma_{2p})=1\) and for primes \(p\equiv 5\pmod 9\), \(\text{rank}_ \mathbb{Q}(\Gamma_{2p^ 2})=1\). (iii) For primes \(p\equiv 7\pmod{24}\), the curve \(y^ 2=x^ 3-p^ 3\) has rank one over \(\mathbb{Q}\). Use of modular functions and class field theory is made and the paper provides excellent examples to the general theory.
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 11G15
Mathematics Subject Classification ID: 14H52
Mathematics Subject Classification ID: 11D25
zbMATH DE Number: 178003
zbMATH Keywords: rational points
zbMATH Keywords: Birch-Swinnerton-Dyer conjecture
zbMATH Keywords: elliptic curves
zbMATH Keywords: complex multiplications
zbMATH Keywords: modular functions
zbMATH Keywords: class field theory
reviewed by: A. R. Rajwade
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 178003