The monodromy problem and the tangential center problem

We consider families of Abelian integrals arising from perturbations of planar Hamiltonian systems. The tangential center-focus problem asks for conditions under which these integrals vanish identically. The problem is closely related to the monodromy problem, which asks when the monodromy of a vanishing cycle generates the whole homology of the level curves of the Hamiltonian. We solve both of these questions for the case in which the Hamiltonian is hyperelliptic. As a by-product, we solve the corresponding problems for the 0-dimensional Abelian integrals defined by Gavrilov and Movasati.