Point counting, division in the Jacobian, and étale cohomology

In this talk, we will give an overview of existing algorithms related to counting points on smooth projective curves over finite fields as well as dividing by integers in their Jacobian, and show how to use such algorithms in order to compute the cohomology of $\pi_1$-modules on such curves. We will explain the part these methods play in point counting algorithms on surfaces, and the work that remains to be done in order to achieve the latter goal with a polynomial-time complexity.