TAILIEUCHUNG - Đề tài " The Hasse principle for pairs of diagonal cubic forms "

By means of the Hardy-Littlewood method, we apply a new mean value theorem for exponential sums to confirm the truth, over the rational numbers, of the Hasse principle for pairs of diagonal cubic forms in thirteen or more variables. 1. Introduction Early work of Lewis [14] and Birch [3], [4], now almost a half-century old, shows that pairs of quite general homogeneous cubic equations possess non-trivial integral solutions whenever the dimension of the corresponding intersection is suitably large (modern refinements have reduced this permissible affine dimension to 826; see [13]). . | Annals of Mathematics The Hasse principle for pairs of diagonal cubic forms By Jorg BrUdern and Trevor D. Wooley Annals of Mathematics 166 2007 865 895 The Hasse principle for pairs of diagonal cubic forms By Jorg Brudern and Trevor D. WooLEy Abstract By means of the Hardy-Littlewood method we apply a new mean value theorem for exponential sums to confirm the truth over the rational numbers of the Hasse principle for pairs of diagonal cubic forms in thirteen or more variables. 1. Introduction Early work of Lewis 14 and Birch 3 4 now almost a half-century old shows that pairs of quite general homogeneous cubic equations possess non-trivial integral solutions whenever the dimension of the corresponding intersection is suitably large modern refinements have reduced this permissible affine dimension to 826 see 13 . When s is a natural number let aj bj 1 j s be fixed rational integers. Then the pioneering work of Davenport and Lewis 12 employs the circle method to show that the pair of simultaneous diagonal cubic equations a1x3 a2x3 . asx3 b1xi b2x2 . bsx3 0 possess a non-trivial solution x G Zs 0 provided only that s 18. Their analytic work was simplified by Cook 10 and enhanced by Vaughan 16 these authors showed that the system necessarily possesses non-trivial integral solutions in the cases s 17 and s 16 respectively. Subject to a local solubility hypothesis a corresponding conclusion was obtained for s 15 by Baker and Brudern 2 and for s 14 by Brudern 5 . Our purpose in this paper is the proof of a similar result that realises the sharpest conclusion attainable by any version of the circle method as currently envisioned even Supported in part by NSF grant DMS-010440. The authors are grateful to the Max Planck Institut in Bonn for its generous hospitality during the period in which this paper was conceived. 866 JORG BRUDERN AND TREVOR D. WOOLEY if one were to be equipped with the most powerful mean value estimates for Weyl sums conjectured to hold. Theorem .

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