Pointless curves of genus three and four

Everett W. Howe, Kristin E. Lauter, and Jaap Top: Pointless curves of genus three and four, pp. 125–141 in: Algebra, Geometry, and Coding Theory (AGCT 2003) (Y. Aubry and G. Lachaud, eds.), Séminaires et Congrès 11, Société Mathématique de France, Paris, 2005, MR 2006g:11125.

(A preprint and an official version are available online, and slides of a talk are available too.)

A curve over a field k is pointless if it has no k-rational points. We show that there exist pointless genus-3 curves over a finite field F_q if and only if either q < 26 or q = 29 or q = 32, and we show that there exist pointless genus-4 curves over a finite field F_q if and only if q < 50.

In fact, for genus-3 curves we prove a little more. We show that there are pointless genus-3 hyperelliptic curves over F_q if and only if q < 26, and that there are pointless plane quartics over F_q if and only if either q < 24 or q = 29 or q = 32.

To prove these results we make use of a number of Magma programs.

Genus3F25.magma. This is the program we use to show that there is exactly one pointless genus-3 curve over F₂₅.
Genus3F27.magma. This is the program we use to show that there are no pointless genus-3 curves over F₂₇.
Genus3Hyperelliptic.magma. This is the program we use to enumerate pointless genus-3 hyperelliptic curves over an arbitrary finite field F_q with q odd and bigger than 7.
Genus4F53.magma. This is the program we use to show that there are no pointless genus-4 curves over F₅₃.
Genus4F59.magma. This is the program we use to show that there are no pointless genus-4 curves over F₅₉.

We also require a list of the isomorphism classes of unimodular quaternary Hermitian forms over the quadratic ring with discriminant -11; such a list is available from the Hermitian lattices web site maintained by Rainer Schulze-Pillot-Zieman. A copy of the relevant file, copied from the math.uni-sb.de server on 22 January 2004, is available locally here. Finally, we include a Magma program examples.magma in which we verify that the curves we present in our paper actually are pointless.