$p$-adic $L$-functions for Hecke characters of totally imaginary fields

We construct $p$-adic $L$-functions interpolating critical $L$-values of algebraic Hecke characters for arbitrary unramified primes $p$ and any totally imaginary field. For non-ordinary primes, the only previously known case was that of imaginary quadratic extensions of $\mathbb{Q}$. One of the main ingredients is a new $p$-adic Fourier theory relating generic fibers of $p$-divisible groups to a general class of character varieties. Combining this with equivariant cohomology classes constructed in a previous paper allows us to construct the $p$-adic $L$-function.