Brumer-Stark Units and Explicit Class Field Theory

Understand

Let $F$ be a totally real field of degree $n$ and $p$ an odd prime.

• We prove the $p$-part of the integral Gross--Stark conjecture for the Brumer--Stark $p$-units living in CM abelian extensions of $F$.
• In previous work, the first author showed that such a result implies an exact $p$-adic analytic formula for these Brumer--Stark units up to a bounded root of unity error, including a ``real multiplication'' analogue of Shimura's celebrated reciprocity law from the theory of Complex Multiplication.
• In this paper we show that the Brumer--Stark units, along with $n-1$ other easily described elements (these are simply square roots of certain elements of $F$) generate the maximal abelian extension of $F$.