i will describe a construction which associates a canonical $p$-adic l-function with a refined cohomological hilbert modular form $(\pi, \alpha)$ under some mild and natural assumptions. the main novelty is that we do not impose any hypothesis of “small slope” or “noncriticality” on the allowable refinements. over $\mathbb{q}$, this result is due to bellaiche. our strategy for dealing with critical refinements is roughly parallel to his, and in particular relies on a careful study of the local geometry of eigenvarieties at classical (but possibly critical) points. this is joint work with john bergdall.