On octonary cubic forms (Q2874670)

This paper gives a conditional proof of the Hasse principle for nonsingular cubic forms defined over the rationals, with 8 (or more) variables. It represents significant progress in our handling of cubic forms via analytic methods.NEWLINENEWLINEGenerally, if \(C(x_1,\ldots,x_n)\) is a nonsingular cubic form with rational integer coefficients, then it was shown by the reviewer [Proc. Lond. Math. Soc. (3) 47, 225--257 (1983; Zbl 0494.10012)] that \(C=0\) always has a non-trivial integer solution as soon as \(n\geq 10\). This was improved by the author of the present paper [J. Reine Angew. Math. 386, 32--98 (1988; Zbl 0641.10019)], who was able to handle the corresponding situation for nonsingular cubics in \(n=9\) variables, for which the Hasse principle was established. Indeed in subsequent papers the smoothness condition was relaxed somewhat.NEWLINENEWLINEThe papers cited above used the circle method with Kloosterman's refinement. Thus one counts solutions of \(C=0\) using a generating function evaluated at rational points \(a/q\), and obtains a saving by averaging non-trivially over the numerators \(a\). In principle one can hope to perform a ``second Kloosterman refinement'', in which one extracts a further saving by averaging non-trivially over the denominators \(q\). The author first demonstrated the feasibility of this with the special cubic form in \(n=6\) variables, formed as a difference of sums of 3 cubes [Acta Math. 157, 49--97 (1986; Zbl 0614.10038)]. The necessarily cancellation as one sums over \(q\) is not currently available unconditionally, but requires some plausible assumptions about the Hasse-Weil \(L\)-functions attached to hyperplane slices of the variety \(C=0\).NEWLINENEWLINEThe present paper extends the above techniques to general nonsingular cubic hypersurfaces \(C=0\). If the method could be pushed through completely it would handle forms in \(n=7\) variables, but the current work contents itself with the case \(n=8\). There are many technical details to be dealt with. Compared to the earliest works on the subject, a major simplification arises from the use of the \(\delta\)-function method of Duke, Friedlander and Iwaniec [\textit{W. Duke} et al., Invent. Math. 112, No. 1, 1--8 (1993; Zbl 0765.11038)], as developed by the reviewer [J. Reine Angew. Math. 481, 149--206 (1996; Zbl 0857.11049)]. This allows one to restrict attention to sums NEWLINE\[NEWLINES(q;\mathbf{c}):=\sum_{\substack{ 1\leq a\leq q\\ (a,q)=1}} \exp\left(2\pi i\frac{aC(\mathbf{x})+\mathbf{c}^T\mathbf{x}}{q}\right).NEWLINE\]NEWLINE One can average over square-free values of \(q\), assuming that the Hasse-Weil \(L\)-function of the variety \(C(\mathbf{x})= \mathbf{c}^T\mathbf{x}=0\) has a suitable analytic continuation to \(\mathbb{C}\), has a standard type of functional equation, and satisfies the relevant Riemann hypothesis. Square-full values of \(q\) are more troublesome, and it is the estimation of these that is the most difficult part of the paper. Indeed the problems here are a major reason for the restriction to 8 rather than 7 variables.