On the assumption of a Riemann hypothesis for certain Hasse-Weil L-functions, it is shewn that a quaternary cubic form f(x) with rational integral coefficients and non-vanishing discriminant represents through integral vectors x almost all integers N having the (necessary) property that the equation f(x)=N is soluble in every p-adic field ℚₚ. The corresponding proposition for quinary forms is established unconditionally.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa8189-1-2016,
author = {C. Hooley},
title = {On the representation of numbers by quaternary and quinary cubic forms: I},
journal = {Acta Arithmetica},
volume = {172},
year = {2016},
pages = {19-39},
zbl = {06574970},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa8189-1-2016}
}
C. Hooley. On the representation of numbers by quaternary and quinary cubic forms: I. Acta Arithmetica, Tome 172 (2016) pp. 19-39. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa8189-1-2016/