On the representation of numbers by quaternary and quinary cubic forms: I
Volume 173 / 2016
Acta Arithmetica 173 (2016), 19-39
MSC: Primary 11D72.
DOI: 10.4064/aa8189-1-2016
Published online: 30 March 2016
Abstract
On the assumption of a Riemann hypothesis for certain Hasse–Weil $L$-functions, it is shewn that a quaternary cubic form $f(\boldsymbol{x})$ with rational integral coefficients and non-vanishing discriminant represents through integral vectors $\boldsymbol{x}$ almost all integers $N$ having the (necessary) property that the equation $f(\boldsymbol{x})=N$ is soluble in every $p$-adic field $\mathbb{Q}_p.$ The corresponding proposition for quinary forms is established unconditionally.
