The Hasse principle for a class of cubic surfaces
type: Item
instance of: scholarly article
title: The Hasse principle for a class of cubic surfaces
zbMATH Open document ID: 0962.11019
author: Samir Siksek
published in: Arab Journal of Mathematical Sciences
publication date: 2001-02-19T00:00:00Z
review text: A classical theorem of \textit{E. S. Selmer} [Math. Scand. 1, 113-119 (1953; Zbl 0051.03202)] asserts that the cubic surface \[ aX^3+ cY^3= bZ^3+ dW^3, \] where \(a,b,c,d\in \mathbb{Z}- \{0\}\) with \(c/a= d/b\), satisfies the Hasse principle. Let \(K\) be a cubic extension of a number field \(k\). In this paper the author generalizes Selmer's theorem by proving that the cubic surface \[ a\text{ Norm}_{K/k} (X+Y\theta)= b\text{ Norm}_{K/k} (Z+W\phi), \] where \(a,b\in k^*\) and \(\theta,\phi\in K\setminus k\), satisfies the Hasse principle.
Mathematics Subject Classification ID: 11D25
Mathematics Subject Classification ID: 14G05
Mathematics Subject Classification ID: 14J20
zbMATH DE Number: 1513043
zbMATH Keywords: cubic surface
zbMATH Keywords: Selmer's theorem
zbMATH Keywords: Hasse principle
reviewed by: Dimitros Poulakis
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1513043