In this paper, the authors study the displacement map associated to small one-parameter polynomial unfoldings of polynomial Hamiltonian vector fields on the plane. The leading term of the displacement map, the generating function \(M(t)\), has an analytic continuation in the complex plane and the real zeros of \(M(t)\) correspond to the limit cycles bifurcating from the periodic orbits of the Hamiltonian flow. The authors give a geometric description of the monodromy group of \(M(t)\), use it to formulate sufficient conditions for \(M(t)\) to satisfy a differential equation of Fuchs or Picard-Fuchs type, and consider some special examples.