Let \(F(\vec x)\), \(\vec x:= (x_1,\dots, x_n)\), be a cubic homogeneous polynomial, let \(F(\vec x)\in\mathbb{Z}[\vec x]\), and let \(X\) be the projective hypersurface in \(\mathbb{P}^{n-1}\) defined by the equation \(F(\vec x)= 0\). The author proves that \(X(\mathbb{Q})\neq\emptyset\) for \(n\geq 14\) and advances the following conjectures:
(i) if \(n\geq 10\), then \(X(\mathbb{Q})\neq\emptyset\); (ii) if \(5\leq n\leq 9\) and the cubic form \(F(\vec x)\) is non-singular, then the hypersurface \(X\) satisfies the Hasse principle; (iii) if \(n\in\{3, 4\}\), then \(X(\mathbb{Q})\neq\emptyset\) as soon as \(X(\mathbb{Q}_p)\neq\emptyset\) for every \(p\)-adic completion \(\mathbb{Q}_p\) and the Brauer-Manin obstruction is empty.
Although the author’s strategy is similar in several ways to that of H. Davenport [Proc. R. Soc. Lond., Ser. A 272, 285–303 (1963; Zbl 0107.04102)], extending the admissible range for \(n\) from Davenport’s \(n\geq 16\) to \(n\geq 14\) requires substantial new ideas. The author supplements Weyl’s inequality with van der Corput’s method combined with a certain averaging process; the rather technical details of that argument can not be described here.