The recent emerging field of synthetic dimension in photonics offers a variety of opportunities for manipulating different internal degrees of freedom of photons such as the spectrum of light. While nonlinear optical effects can be incorporated into these photonic systems with synthetic dimensions, these nonlinear effects typically result in long-range interactions along the frequency axis. Thus, it has been difficult to use the synthetic dimension concept to study a large class of Hamiltonians that involves local interactions. Here we show that a Hamiltonian that is locally interacting along the synthetic dimension can be achieved in a dynamically modulated ring resonator incorporating ${\chi }^{(3)}$ nonlinearity, provided that the group velocity dispersion of the waveguide forming the ring is specifically designed. As a demonstration we numerically implement a Bose–Hubbard model and explore photon blockade effect in the synthetic frequency space. Our work opens new possibilities for studying fundamental many-body physics in the synthetic space in photonics, with potential applications in optical quantum communication and quantum computation.