On the density of modular points in universal deformation spaces

Based on comparison theorems for Hecke algebras and universal deformation rings with strong restrictions at the critical prime l, as provided by the results of Wiles, Taylor, Diamond, et al., we prove under rather general conditions that the corresponding universal deformation spaces with no restrictions at l can be identified with certain Hecke algebras of l-adic modular forms as conjectured by Gouvêa, thus generalizing previous work of Gouvêa and Mazur. Along the way, we show that the universal deformation spaces we consider are complete intersections, flat over [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /]_l of relative dimension three, in which the modular points form a Zariski dense subset. Furthermore the fibers above [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /]_l of these spaces are generically smooth.