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The Hasse principle for pairs of diagonal cubic forms. (English) Zbl 1171.11053

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 H. Davenport and 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 T. Kawada [Ann. Sci. École Norm. Sup. 34, No. 4, 471–501 (2001; Zbl 1020.11062)], for example.

MSC:

11P55 Applications of the Hardy-Littlewood method
11E76 Forms of degree higher than two
14G05 Rational points
11P05 Waring’s problem and variants
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