Described herein is a hydrocracking and/or hydrotreatment process which uses a catalyst comprising an active phase containing at least one hydrogenating/dehydrogenating component, and a support comprising at least one dealuminated zeolite Y having an overall initial atomic ratio of silicon to aluminum between 2.5 and 20, an initial weight fraction of extra-lattice aluminum atoms greater than 10%, an initial mesopore volume greater than 0.07 ml.g−1, and an initial crystal lattice parameter a0 between 24.38 Å and 24.30 Å. The zeolite Y is modified by a basic treatment stage, and at least one thermal treatment stage.