The Rubin–Stark conjecture for a special class of function field extensions

Abstract

We prove a strong form of the Brumer–Stark Conjecture and, as a consequence, a strong form of Rubin's integral refinement of the abelian Stark Conjecture, for a large class of abelian extensions of an arbitrary characteristic p global field k. This class includes all the abelian extensions K / k contained in the compositum k p ∞ ≔ k p · k ∞ of the maximal pro- p abelian extension k p / k and the maximal constant field extension k ∞ / k of k, which happens to sit inside the maximal abelian extension k ab of k with a quasi-finite index. This way, we extend the results obtained by the present author in (Comp. Math. 116 (1999) 321–367).