Counting rational points on smooth cyclic covers
Number Theory
2011-09-08 v1
Abstract
A conjecture of Serre concerns the number of rational points of bounded height on a finite cover of projective space P^{n-1}. In this paper, we achieve Serre's conjecture in the special case of smooth cyclic covers of any degree when n is at least 10, and surpass it for covers of degree 3 or higher 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.
Cite
@article{arxiv.1109.1455,
title = {Counting rational points on smooth cyclic covers},
author = {D. R. Heath-Brown and Lillian B. Pierce},
journal= {arXiv preprint arXiv:1109.1455},
year = {2011}
}