While point counting algorithms on algebraic curves over finite fields have been around for decades and seen considerable progress in the meantime, the question of counting points on algebraic surfaces without additional structure seems much more challenging. In this talk, we will present results concerning the classical l-adic method for surfaces fibered over the projective line, which was conjectured by Edixhoven and Couveignes to admit a polynomial-time complexity. This method boils down to computing the étale cohomology of a constructible sheaf: we will describe how to compute this cohomology once such a sheaf has been given, and outline a possible strategy that might allow to compute a usable description of this particular sheaf.