Curves of fixed gonality with many rational points
Floris Vermeulen

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 and an odd prime power we show that for every large genus there exists a non-singular curve defined over of genus and gonality and with exactly -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.
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Meromorphic and Entire Functions · Analytic Number Theory Research
