Let C be a smooth projective geometrically irreducible algebraic curve of genus g over a field k. The Jacobian variety J of C is a g-dimensional algebraic group that parametrizes the degree zero divisors on C, up to linear equivalence. Khuri-Makdisi showed that the basic arithmetic in J can be realized in an asymptotic complexity of O(g^3+ε) field operations in k. Denote by F = k(C) the function field of the curve. Then the elements of J can be identified with divisor classes [D] of the function field F/k where D can be represented by a lattice L_D over the polynomial ring k[t]. In fact, the class [D] can be parametrized in terms of invariants of the lattice L_D. The basic arithmetic (addition and inversion) in J can then be realized asymptotically in O(g³) field operations in k and beats the one of Khuri-Makdisi. Under reasonable assumptions the runtime can be reduced to O(g²) field operations.