(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
**F**_{25}. - Genus3F27.magma. This is the program
we use to show that there are no pointless genus-3 curves
over
**F**_{27}. - 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
**F**_{53}. - Genus4F59.magma. This is the program
we use to show that there are no pointless genus-4 curves
over
**F**_{59}.