On the Bombieri-Pila Method Over Function Fields

TL;DR

This paper extends the Bombieri-Pila method to function fields of genus 0, enabling improved bounds on lattice points and applications to counting elliptic curves within specific coefficient bounds.

Contribution

The paper generalizes the Bombieri-Pila method from number fields to genus 0 function fields, providing new tools for counting elliptic curves with bounded coefficients.

Findings

Extended Bombieri-Pila bounds to genus 0 function fields

Applied the method to count elliptic curves in a coefficient box

Provided new estimates for lattice points in algebraic settings

Abstract

E. Bombieri and J. Pila introduced a method for bounding the number of integral lattice points that belong to a given arc under several assumptions. In this paper we generalize the Bombieri-Pila method to the case of function fields of genus 0 in one variable. We then apply the result to counting the number of elliptic curves contain in an isomorphism class and with coefficients in a box.