Cubic hypersurfaces and a version of the circle method for number fields (Q398990)

This paper achieves two significant goals. The first is to develop a version of the circle method over number fields, using an extension of the ``Delta method'' of \textit{W. Duke} et al. [Invent. Math. 112, No. 1, 1--8 (1993; Zbl 0765.11038)]. In this way the paper provides, for the first time, a Kloosterman refinement for the circle method over number fields. The second achievement is to show that a nonsingular cubic form in 10 or more variables over a number field always has a nontrivial zero over the number field. Equivalently, a smooth projective cubic hypersurface of dimension at least 8, defined over a number field, always has a point over its field of definition. Previously \textit{C. M. Skinner} [Duke Math. J. 75, No. 2, 409--466 (1994; Zbl 0848.14009)] had handled nonsingular cubic forms in 13 or more variables, over arbitrary number fields, but further progress was thwarted by the lack of a Kloosterman refinement.NEWLINENEWLINEFor the first of these themes the authors follow the reviewer's paper [J. Reine Angew. Math. 481, 149--206 (1996; Zbl 0857.11049)], in which a Delta-method version of the circle method is developed for \(\mathbb{Q}\). There are a number of obstacles to be overcome in extending the reviewer's work. In particular the present paper avoids the use of the Euler--Maclaurin summation formula by appealing to a Mellin transform involving the Dedekind zeta-function.NEWLINENEWLINEHaving developed their formulation of the circle method with Kloosterman refinement, the authors follow a second paper by the reviewer [Proc. Lond. Math. Soc. (3) 47, 225--257 (1983; Zbl 0494.10012)] to handle nonsingular cubic forms. This requires the estimation of smoothly weighted multi-dimensional cubic exponential integrals. The authors are able to re-use some of Skinner's work here. Similarly there are cubic exponential sums to consider, for which Deligne's bounds are applied when the modulus is square-free. As in the reviewer's work on the rational case, square-full moduli are dealt with by treating a suitable average of sums, which suffices for the application.NEWLINENEWLINEThe authors mention briefly that their methods can be applied to equations \(Q(x_1,\ldots,x_n)=a\) over the ring of integers of a number field, in which \(Q\) is a non-singular quadratic form. When \(n\geq 4\) they are able to give the correct asymptotics for the number of solutions. This was already known for \(n\geq 5\) (\textit{C. M. Skinner} [Compos. Math. 106, No. 1, 11--29 (1997; Zbl 0892.11014)]), but the case \(n=4\) seems to require a Kloosterman refinement, as in the present paper.