Brumer–Stark units and explicit class field theory

Abstract

Let F be a totally real field of degree n, and let p be 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. We therefore obtain an unconditional construction of the maximal abelian extension of any totally real field, albeit one that involves p-adic integration for infinitely many primes p.

Our method of proof of the integral Gross–Stark conjecture is a generalization of our previous work on the Brumer–Stark conjecture. We apply Ribet’s method in the context of group ring valued Hilbert modular forms. A key new construction here is the definition of a Galois module ${\nabla }_L$ that incorporates an integral version of the Greenberg–Stevens L-invariant into the theory of Ritter–Weiss modules. This allows for the reinterpretation of Gross’s conjecture as the vanishing of the Fitting ideal of ${\nabla }_L$. This vanishing is obtained by constructing a quotient of ${\nabla }_L$ whose Fitting ideal vanishes using the Galois representations associated to cuspidal Hilbert modular forms.