On a generalization of the rank one Rubin–Stark conjecture

In this paper we study further the extended abelian rank one Stark conjecture contained in Emmons and Popescu (2009) [4], and Erickson (2009) [5], . We formulate a stronger question (Question 4.2) which seems easier to investigate both theoretically and computationally. Question 4.2 includes a generalization of the Brumer–Stark conjecture on annihilation of class groups (see Question 4.7). We link it with a conjecture of Gross (contained in Gross (1988) [6]), and in the process find some new integrality properties of the Stickelberger element (Theorem 4.30). Finally, we provide some numerical examples with base field Q for which Question 4.2, and thus the extended abelian rank one Stark conjecture, have an affirmative answer.