  We construct examples of p-adic L-functions over universal deformation spaces
for GL(2). We formulate a conjecture predicting that the natural parameter
spaces for p-adic L-functions are not the usual eigenvarieties (parametrising
nearly-ordinary families of automorphic representations), but other, larger
spaces depending on a choice of a parabolic subgroup, which we call "big
parabolic eigenvarieties".
