Rational points on a class of cubic hypersurfaces

Abstract Let r ⩾ 3 r 3 be an integer and 𝑄 any positive definite quadratic form in 𝑟 variables. We establish asymptotic formulae with power-saving error terms for the number of rational points of bounded height on singular hypersurfaces S Q Sₐ defined by x 3 = Q ⁢ (y 1, …, y r) ⁢ z x^3=Q (y₁, , yₑ) z. This confirms Manin’s conjecture for any S Q Sₐ. Our proof is based on analytic methods, and uses some estimates for character sums and moments of 𝐿-functions. In particular, one of the ingredients is Siegel’s mass formula in the argument for the case r = 3 r=3.