  We study the problem of lifting curves from finite fields to number fields in
a genus and gonality preserving way. More precisely, we sketch how this can be
done efficiently for curves of gonality at most four, with an in-depth
treatment of curves of genus at most five over finite fields of odd
characteristic, including an implementation in Magma. We then use such a lift
as input to an algorithm due to the second author for computing zeta functions
of curves over finite fields using $p$-adic cohomology.
