On the Hasse principle for quartic hypersurfaces (Q2336102)

Let \(F(X_1,\ldots,X_n)\in\mathbb{Z}[X_1,\ldots,X_n]\) be a quartic form defining a projective variety \(V\) with a smooth point over every completion of \(\mathbb{Q}\). It is shown in this paper that the associated point counting function satisfies \(N(V;B)\gg B^{n-4}\) as soon as \(n\ge \dim(\mathrm{sing}(V))+31\). In particular, this condition on \(n\) suffices for the Hasse principle. Previous results in this context had 31 replaced by 50 [\textit{B. J. Birch}, Proc. R. Soc. Lond., Ser. A 265, 245--263 (1962; Zbl 0103.03102)], by 42 [\textit{T. D. Browning} and \textit{D. R. Heath-Brown}, J. Reine Angew. Math. 629, 37--88 (2009; Zbl 1169.11027)] and by 41 [\textit{M. A. Hanselmann}, Rational points on quartic hypersurfaces. München: Univ. München, Fakultät für Mathematik, Informatik und Statistik (Diss.) (2012; Zbl 1296.14001)]. Thus the reduction to 31 is quite substantial. Birch's paper used the Hardy-Littlewood circle method and bounded the quartic exponential sums which arise by using three Weyl differencing steps. The subsequent improvements use van der Corput differencing, followed by an application of Poisson summation and estimates for complete exponential sums. The present paper combines these ideas with the Kloosterman refinement. Indeed this seems to be the first time that anyone has been able to apply the Kloosterman refinement for forms of degree more than 3.