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.

Statement of the conjecture

Let Template:Math be an abelian extension of global fields, and let Template:Math be a set of places of Template:Math containing the Archimedean places and the prime ideals that ramify in Template:Math. The Template:Math-imprimitive equivariant Artin L-function Template:Math is obtained from the usual equivariant Artin L-function by removing the Euler factors corresponding to the primes in Template:Math 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 Template:Math where Template:Math is the Galois group of Template:Math. It is analytic on the entire plane, excepting a lone simple pole at Template:Math.

Let Template:Math be the group of roots of unity in Template:Math. The group Template:Math acts on Template:Math; let Template:Math be the annihilator of Template:Math as a Template:Math-module. An important theorem, first proved by C. L. Siegel and later independently by Takuro Shintani, states that Template:Math is actually in Template:Math. A deeper theorem, proved independently by Pierre Deligne and Ken Ribet, Daniel Barsky, and Pierrette Cassou-Noguès, states that Template:Math is in Template:Math. In particular, Template:Math is in Template:Math, where Template:Math is the cardinality of Template:Math.

The ideal class group of Template:Math is a [[G-module|Template:Math-module]]. From the above discussion, we can let Template:Math act on it. The Brumer–Stark conjecture says the following:

Brumer–Stark Conjecture. For each nonzero fractional ideal $𝔞$ of Template:Math, there is an "anti-unit" Template:Math such that

1. ${𝔞}^{W\theta (0)}=(\epsilon ).$
2. The extension $K\left({\epsilon }^{\frac{1}{W}}\right)/k$ is abelian.

The first part of this conjecture is due to Armand Brumer, and Harold Stark originally suggested that the second condition might hold. The conjecture was first stated in published form by John Tate.

The term "anti-unit" refers to the condition that Template:Math is required to be 1 for each Archimedean place Template:Math.

Progress

The Brumer Stark conjecture is known to be true for extensions Template:Math where

• Template:Math is cyclotomic: this follows from Stickelberger's theorem
• Template:Math is abelian over Template:Math
• Template:Math is a quadratic extension
• Template:Math is a biquadratic extension

In 2020, Dasgupta and Kakde proved the Brumer–Stark conjecture away from the prime 2. In 2023, a full proof of the conjecture over Template:Math has been announced.

Function field analogue

The analogous statement in the function field case is known to be true, having been proved by John Tate and Pierre Deligne in 1984, with a different proof by David Hayes in 1985.