We discuss a rather general algorithm for point counting on a (nice) curve over a finite field of characteristic p, assuming a lift to the valuation ring in a finite extension of Q_p has been given. Our method for computing the action of Frobenius is based on computing cup products in cohomology, not on explicitly rewriting 1-forms using exact 1-forms. These cup products can be computed locally, and the same holds for the local expansions of a lift of Frobenius that is determined uniquely by some global equations. This is joint work with Amnon Besser, Pengju Guan, and Muxi Li.