Counting rational points on smooth cyclic covers

TL;DR

This paper proves Serre's conjecture on the number of rational points of bounded height for smooth cyclic covers of projective space in high dimensions, using novel bounds involving power sieves and van der Corput's method.

Contribution

It establishes Serre's conjecture for smooth cyclic covers of any degree when dimension is at least 10, and improves bounds for degrees 3 or higher when dimension exceeds 10.

Findings

Proves Serre's conjecture for smooth cyclic covers in high dimensions.

Develops a new bound for perfect r-th power values of polynomials.

Combines power sieve and van der Corput's method for improved estimates.

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.