Brumer–Stark conjecture

The Brumer–Stark conjecture is a conjecture in algebraic number theory giving a rough generalization of both the analytic class number formula for Dedekind zeta functions, and also of Stickelberger's theorem about the factorization of Gauss sums. It is named after Armand Brumer and Harold Stark.

It arises as a special case (abelian and first-order) of Stark's conjecture, when the place that splits completely in the extension is finite. There are very few cases where the conjecture is known to be valid. Its importance arises, for instance, from its connection with Hilbert's twelfth problem.

Let K/k be an abelian extension of global fields, and let S be a set of places of k containing the Archimedean places and the prime ideals that ramify in K/k. The S-imprimitive equivariant Artin L-function θ(s) is obtained from the usual equivariant Artin L-function by removing the Euler factors corresponding to the primes in S from the Artin L-functions from which the equivariant function is built. It is a function on the complex numbers taking values in the complex group ring C[G] where G is the Galois group of K/k. It is analytic on the entire plane, excepting a lone simple pole at s = 1.

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