On the density of modular points in universal deformation spaces

Based on comparison theorems for Hecke algebras and universal deformation r ings with strong restrictions at the critical prime l, as provided by the r esults of Wiles, Taylor, Diamond, et al., we prove under rather general con ditions that the corresponding universal deformation spaces with no restric tions at l can be identified with certain Hecke algebras of l-acid modular forms as conjectured by Gouvea, thus generalizing previous work of Gouvea a nd Mazur. Along the way, we show that the universal deformation spaces we c onsider are complete intersections, flat over Z(l) of relative dimension th ree, in which the modular points form a Zariski dense subset. Furthermore t he fibers above Q(l) of these spaces are generically smooth.