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

TL;DR

This paper constructs $p$-adic $L$-functions for algebraic Hecke characters over totally imaginary fields, extending previous work to arbitrary unramified primes and general fields using new $p$-adic Fourier theory.

Contribution

It introduces a novel $p$-adic Fourier theory and combines it with equivariant cohomology to construct $p$-adic $L$-functions for a broad class of fields and primes.

Findings

Successfully interpolates critical $L$-values for arbitrary unramified primes

Extends the construction of $p$-adic $L$-functions beyond imaginary quadratic fields

Provides new tools for studying $p$-adic properties of Hecke characters

Abstract

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.