The Hasse principle for pairs of diagonal cubic forms (Q2482876)

This paper shows that if the coefficients \(a_i,b_i\) are arbitrary integers, and if \(s\geq 13\), the simultaneous equations \[ \sum^s_{i=1} a_ix_i^3=\sum^s_{i=1} b_ix_i^3=0 \] will have a nontrivial integer solution if and only if they have a nontrivial solution in each \(p\)-adic field. In fact this latter condition holds automatically unless \(p=7\). The bound on \(s\) has been successively reduced by various authors, beginning with \textit{H. Davenport} and \textit{D. J. Lewis} [Philos. Trans. R. Soc. Lond., Ser. A 261, 97--136 (1966; Zbl 0227.10038)], who showed that \(s\geq 18\) sufficed. The result includes the Hasse Principle for a single diagonal form in 7 variables as a very special case. Indeed any improvement of the present result to systems in 12 variables would imply the Hasse Principle for a single form in 6 variables -- a result apparently well outside our current capabilities. The first key ingredient in the proof is the second author's estimate [Invent. Math. 122, 421--451 (1995; Zbl 0851.11055)] \[ \int^1_0| h(\alpha)|^6\,d\alpha\ll P^{3+1/4-\delta}, \] where the cubic Weyl sum \[ h(\alpha)=\sum_ne(\alpha n^3) \] is restricted to suitably smooth \(n\leq P\). This is used in establishing a mixed 12-th power moment estimate \[ \int^1_0\int^1_0| h(a\alpha)|^5| h(b \beta)|^5| h(c\alpha+ d\beta)|^2\,d\alpha\, d\beta\ll P^{6+1/4 -\delta}, \] valid for any non-zero integers \(a,b,c,d\). This is then used, via Hölder's inequality, to handle the minor arcs in the two-dimensional circle method. The proof of the mixed 12-th power moment estimate is the most novel part of the argument, and is accomplished using ``slim exceptional set'' technology, as developed in the authors' paper with \textit{T. Kawada} [Ann. Sci. École Norm. Sup. 34, No. 4, 471--501 (2001; Zbl 1020.11062)], for example.