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

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 kp∞≔kp·k∞ of the maximal pro-p abelian extension kp/k and the maximal constant field extension k∞/k of k, which happens to sit inside the maximal abelian extension kab 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).