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.
An important theorem, first proved by C. L. Siegel and later independently by Takuro Shintani, states that θ(0) is actually in Q[G].
A deeper theorem, proved independently by Pierre Deligne and Ken Ribet, Daniel Barsky, and Pierrette Cassou-Noguès, states that Aθ(0) is in Z[G].
of K, there is an "anti-unit" ε such that The first part of this conjecture is due to Armand Brumer, and Harold Stark originally suggested that the second condition might hold.
[7] The analogous statement in the function field case is known to be true, having been proved by John Tate and Pierre Deligne in 1984,[8] with a different proof by David Hayes in 1985.