Counting points on cubic surfaces, I

Slater, J.B. and Swinnerton-Dyer, P. (1998) Counting points on cubic surfaces, I. Asterisque (251). pp. 1-12. ISSN 0303-1179. (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided)

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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
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