The Hasse principle for a class of cubic surfaces
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.
scientific article; zbMATH DE number 1513043