On the maximum gonality of a curve over a finite field

Faber, Xander; Grantham, Jon; Howe, Everett W.

doi:10.2140/ant.2025.19.1637

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1944-7833 (online)

ISSN 1937-0652 (print)

Author index

To appear

Other MSP journals

On the maximum gonality of a curve over a finite field

Xander Faber, Jon Grantham and Everett W. Howe

Vol. 19 (2025), No. 8, 1637–1662

DOI: 10.2140/ant.2025.19.1637

Abstract

The gonality of a smooth geometrically connected curve over a field $k$ is the smallest degree of a nonconstant $k$ -morphism from the curve to the projective line. In general, the gonality of a curve of genus $g \geq 2$ is at most $2 g - 2$ . Over finite fields, a result of F. K. Schmidt from the 1930s can be used to prove that the gonality is at most $g + 1$ . Via a mixture of geometry and computation, we improve this bound: for a curve of genus $g \geq 5$ over a finite field, the gonality is at most $g$ . For genus $g = 3$ and $g = 4$ , the same result holds with exactly $217$ exceptions: there are two curves of genus $4$ and gonality $5$ , and $215$ curves of genus $3$ and gonality $4$ . The genus- $4$ examples were found in other papers, and we reproduce their equations here; in supplementary material, we provide equations for the genus- $3$ examples.

Keywords

curves over finite fields, gonality

Mathematical Subject Classification

Primary: 11G20, 14H51

Secondary: 14Q05

Milestones

Received: 4 March 2024

Revised: 30 July 2024

Accepted: 13 September 2024

Published: 12 June 2025

Authors

Xander Faber
Center for Computing Sciences Institute for Defense Analyses Bowie, MD United States

Jon Grantham
Center for Computing Sciences Institute for Defense Analyses Bowie, MD United States

Everett W. Howe
San Diego, CA United States

Open Access made possible by participating institutions via Subscribe to Open.

Vol. 19, No. 8, 2025