Geometric generalizations of the square sieve, with an application to cyclic covers

TL;DR

This paper develops a geometric sieve method to estimate the number of rational points of bounded height on cyclic covers over function fields, extending classical conjectures to a new geometric context.

Contribution

It introduces a new geometric sieve technique applicable to cyclic covers over global function fields, providing bounds on rational points that generalize Serre's conjecture.

Findings

Provides bounds for points on cyclic covers over $ ext{F}_q(T)$

Generalizes polynomial sieve to a geometric setting

Extends Serre's conjecture to function fields

Abstract

We formulate a general problem: given projective schemes $\mathbb{Y}$ and $\mathbb{X}$ over a global field $K$ and a $K$-morphism $\eta$ from $\mathbb{Y}$ to $\mathbb{X}$ of finite degree, how many points in $\mathbb{X}(K)$ of height at most $B$ have a pre-image under $\eta$ in $\mathbb{Y}(K)$? This problem is inspired by a well-known conjecture of Serre on quantitative upper bounds for the number of points of bounded height on an irreducible projective variety defined over a number field. We give a non-trivial answer to the general problem when $K=\mathbb{F}_q(T)$ and $\mathbb{Y}$ is a prime degree cyclic cover of $\mathbb{X}=\mathbb{P}_{K}^n$. Our tool is a new geometric sieve, which generalizes the polynomial sieve to a geometric setting over global function fields.