Let C be a projective smooth curve of genus g over a field k. The Picard group of C over k, consisting of k-rational divisors up to linear equivalence, is analogous to the ideal class group of a number field. In this talk, I will describe "geometric" algorithms for the group arithmetic in Pic⁰ that boil down to linear algebra in vector spaces of dimension O(g log g). Using fast linear algebra, this yields a complexity of O(g^2.376) field operations in k per group operation in the Picard group. In comparison, existing "arithmetic" algorithms based on the analogy with ideal class groups have a complexity of O(g⁴) for general curves of genus g, but attain O(g²) if one restricts to special classes of curves.