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 Fq
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 Fq 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
Fq if and only if q < 26, and that
there are pointless plane quartics over Fq
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 Fq 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.