Asymptotically fast group operations on Jacobians of general curves

Let C be a curve of genus g over a field k. We describe probabilistic algorithms for addition and inversion of the classes of rational divisors in the Jacobian of C. After a precomputation, which is done only once for the curve C, the algorithms use only linear algebra in vector spaces of dimension at most O(g log g), and so take O(g3+ ) field operations in k, using Gaussian elimination. Using fast algorithms for the linear algebra, one can improve this time to O(g2.376). This represents a significant improvement over the previous record ofO(g4) field operations (also after a precomputation) for general curves of genus g.