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The circle method and diagonal cubic forms

Abstract:
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+ε) zeros up to height P, for any ε > 0. For four variables we show that there are O(P3/2+ε) such zeros, excluding any that lie on rational lines in the corresponding surface.
Publication status:
Published

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Publisher copy:
10.1098/rsta.1998.0181

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Institution:
University of Oxford
Department:
Oxford, MPLS, Mathematical Inst
Role:
Author
Journal:
PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES
Volume:
356
Issue:
1738
Pages:
673-699
Publication date:
1998-03-15
DOI:
EISSN:
1471-2962
ISSN:
1364-503X
URN:
uuid:cbb63efe-f7e8-4d01-88a8-27837fc80e83
Source identifiers:
15477
Local pid:
pubs:15477

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