The circle method and diagonal cubic forms

被引：30
|
作者：
Heath-Brown, DR [1 ]
机构：
[1] Univ Oxford Magdalen Coll, Oxford OX1 4AU, England
来源：
PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES | 1998年 / 356卷 / 1738期
关键词：
cubic surface; Hasse-Weil L-function; rational lines; Hardy-Littlewood method; rational points; sum of cubes;
D O I：
10.1098/rsta.1998.0181
中图分类号：
O [数理科学和化学]; P [天文学、地球科学]; Q [生物科学]; N [自然科学总论];
学科分类号：
07 ; 0710 ; 09 ;
摘要：
We use the Hardy-Littlewood circle method, in the form developed by Heath-Brown in 1996, to investigate the number of integer zeros of diagonal cubic forms. The results are subject to unproved hypotheses concerning certain Hasse-Weil L-functions. For six variables we show that there are O(P3+epsilon) zeros up to height P, for any epsilon > 0. For four variables we show that there are O(P3/2+epsilon) such zeros, excluding any that lie on rational lines in the corresponding surface.
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页码：673 / 699
页数：27
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