Slater, J.B. and Swinnerton-Dyer, P.
(1998)
*
Counting points on cubic surfaces, I.
*
Asterisque
(251).
pp. 1-12.
ISSN 0303-1179.
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## Abstract

Let V be a nonsingular cubic surface defined over Q, let U be the open subset of V obtained by deleting the 27 lines, and denote by N(U, H) the number of rational points in U of height less than H. Manin has conjectured that if V(Q) is not empty then (1) N(U, H) = C1H(log H)(r-1)(1 + o(1)) for some C-1 > 0, where r is the rank of NS(V/Q), the Neron-Severi group of V over Q. In this note we consider the special case when V contains two rational skew lines; and we prove that for some C-2 > 0 and all large enough H, N(U, H) > C2H(log H)(r-1) This is the one-sided estimate corresponding toll). It seems probable that the arguments in this paper could be modified to prove the corresponding result when V contains two skew lines conjugate over Q and each defined over a quadratic extension of Q but we have not attempted to write out the details.

Item Type: | Article |
---|---|

Uncontrolled keywords: | cubic surfaces; Manin conjecture |

Subjects: | Q Science > QA Mathematics (inc Computing science) > QA 75 Electronic computers. Computer science Q Science > QA Mathematics (inc Computing science) |

Divisions: | Faculties > Science Technology and Medical Studies > School of Computing |

Depositing User: | Tara Puri |

Date Deposited: | 02 Jul 2009 09:06 |

Last Modified: | 02 Jul 2009 09:06 |

Resource URI: | https://kar.kent.ac.uk/id/eprint/17146 (The current URI for this page, for reference purposes) |

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