Curves of fixed gonality with many rational points

TL;DR

The paper constructs non-singular algebraic curves over finite fields with fixed gonality and maximal rational points, confirming a recent conjecture and expanding understanding of rational point distributions on curves.

Contribution

It proves the existence of curves with fixed gonality and maximal rational points for large genus, using toric surface methods and Poonen's polynomial squarefree values work.

Findings

Existence of curves with fixed gonality and maximal rational points for large genus

Construction methods based on toric surfaces and polynomial squarefree values

Confirmation of a recent conjecture by Faber and Grantham

Abstract

Given an integer $\gamma\geq 2$ and an odd prime power $q$ we show that for every large genus $g$ there exists a non-singular curve $C$ defined over $\mathbb{F}_q$ of genus $g$ and gonality $\gamma$ and with exactly $\gamma(q+1)$ $\mathbb{F}_q$-rational points. This is the maximal number of rational points possible. This answers a recent conjecture by Faber--Grantham. Our methods are based on curves on toric surfaces and Poonen's work on squarefree values of polynomials.