BRUMER–STARK UNITS AND EXPLICIT CLASS FIELD THEORY

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 rL 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 rL. This vanishing is obtained by constructing a quotient of rL whose Fitting ideal vanishes using the Galois representations associated to cuspidal Hilbert modular forms.

Duke Scholars

Author Samit Dasgupta Mathematics