I will present a randomised algorithm to compute the local zeta function of a fixed smooth, projective surface over the rationals, at any large prime p of good reduction. The runtime of the algorithm is polynomial in log p, answering a question of Couveignes and Edixhoven. The main ingredient is to explicitly compute cocycles associated to a Lefschetz pencil on the surface. This is based on joint work with Nitin Saxena.