  We study a quantum Hamiltonian that models an two--dimensional array of
Josephson junctions with short range Josephson couplings, (given by the
Josephson energy E_J and charging energiey E_C due to the small capacitance of
the junctions. We include the effects from both the self-C0 and the junction-C1
capacitances in the presence of external magnetic flux f=fi/fi0 as well as
uniform background of charges q_{x}. We derive an effective quantum non--linear
sigma-model for the array Hamiltonian which enables us a non mean--field
treatment of the zero--temperature phase transition scenario. We calculate the
ground--state phase diagram, analytically deriving E_J^{crit}(E_C,q_x,f) for
several rational fluxes f=0,1/2,1/3,1/4 and 1/6 that improves upon previous
theoretical treatments based on mean--field approximations.
