On octonary cubic forms

Abstract

Let f ( x 1 , … , x 8 ) be a cubic form in eight variables with rational integral coefficients and non-zero discriminant. Then, assuming a Riemann hypothesis for certain Hasse–Weil L-functions, we prove that the indeterminate equation f ( x 1 , … , x 8 ) = 0 has a non-zero solution provided that the form satisfy the necessary condition that it have a non-trivial zero in every p-adic field Q p . This extends the earlier unconditional results due to Heath–Brown and the author for cubic forms in ten and nine variables.