Curves of fixed gonality with many rational points

Abstract

Given an integer $\gamma \ge 2$ 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.

12 pages. The proof doesn't work in characteristic 2, so the theorem and proof have been updated

Cited by in corpus (1)

• Binary Curves of small fixed genus and gonality with many rational points