Over the past years I have developed an algorithm for computing zeta functions of algebraic curves over finite fields using p-adic cohomology. However, the input to this algorithm is not the curve itself, but rather a lift of it to characteristic zero, together with a map to the projective line (satisfying certain technical conditions). Moreover, the running time of the algorithm depends a lot on the degree of the map. The main remaining question is therefore how to lift a curve to characteristic zero in such a way that it admits a map to the projective line of lowest possible degree. In joint work with Wouter Castryck (https://arxiv.org/abs/1605.02162) we answer this question for curves of genus up to and including five. Although the talk will involve some classical algebraic geometry, we will not assume the audience to be familiar with this.