Geodesic
On the representation of numbers by quaternary and quinary cubic forms: I
C. Hooley 1
1 School of Mathematics Cardiff University Senghennydd Road Cardiff CF24 4AG, United Kingdom
Acta Arithmetica, Tome 173 (2016) no. 1, pp. 19-39 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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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.
DOI : 10.4064/aa8189-1-2016
Keywords: assumption riemann hypothesis certain hasse weil l functions shewn quaternary cubic form boldsymbol rational integral coefficients non vanishing discriminant represents through integral vectors boldsymbol almost integers having necessary property equation boldsymbol soluble every p adic field mathbb corresponding proposition quinary forms established unconditionally

C. Hooley  1

1 School of Mathematics Cardiff University Senghennydd Road Cardiff CF24 4AG, United Kingdom 
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C. Hooley. On the representation of numbers by quaternary and quinary cubic forms: I. Acta Arithmetica, Tome 173 (2016) no. 1, pp. 19-39. doi: 10.4064/aa8189-1-2016

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