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On the representation of numbers by quaternary and quinary cubic forms: I

Volume 173 / 2016

C. Hooley 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.

Authors

  • C. HooleySchool of Mathematics
    Cardiff University
    Senghennydd Road
    Cardiff CF24 4AG, United Kingdom
    e-mail

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