We construct p-adic L-functions associated with p-refined cohomological
cuspidal Hilbert modular forms over any totally real field under a mild
hypothesis. Our construction is canonical, varies naturally in p-adic
families, and does not require any small slope or non-criticality assumptions
on the p-refinement. The main new ingredients are an adelic definition of a
canonical map from overconvergent cohomology to a space of locally analytic
distributions on the relevant Galois group and a smoothness theorem for certain
eigenvarieties at critically refined points.Comment: 88 page