Counting rational points on smooth cyclic covers

Journal Article

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. © 2012 Elsevier Inc.

Full Text

Duke Authors

Cited Authors

  • Heath-Brown, DR; Pierce, LB

Published Date

  • 2012

Published In

Volume / Issue

  • 132 / 8

Start / End Page

  • 1741 - 1757

International Standard Serial Number (ISSN)

  • 0022-314X

Digital Object Identifier (DOI)

  • 10.1016/j.jnt.2012.02.010