Counting rational points on smooth cyclic covers

JOURNAL OF NUMBER THEORY

卷:132

Article

Heath-Brown, D. R.¹  Pierce, Lillian B.¹ 

[1] Math Inst, Oxford OX1 3LB, England

关键词: Rational points;    Cyclic covers;    Power sieve;    Character sums;   

DOI  :  10.1016/j.jnt.2012.02.010

来源: Elsevier

【 摘 要 】

A conjecture of Serre concerns the number of rational points of bounded height on a finite cover of projective space Pn-1. In this paper, we achieve Serre's conjecture in the special case of smooth cyclic covers of any degree when n >= 10, and surpass it for covers of degree r >= 3 when n > 10. This is achieved by a new bound for the number of perfect r-th power values of a polynomial with nonsingular leading form, obtained via a combination of an r-th power sieve and the q-analogue of van der Corput's method. (C) 2012 Elsevier Inc. All rights reserved.

【 授权许可】

Free