UC San Diego

This thesis is concerned with the extended abelian rank one Stark conjecture stated for the first time in Erickson's thesis. Here, we investigate it in more depth then has been done so far. We formulate a stronger question which seems easier to investigate both theoretically and computationally. This question includes a generalization of the Brumer-Stark conjecture on annihilation of class groups. We link it with a conjecture of Gross and in the process find some new integrality properties of the Stickelberger element. We provide some numerical examples, when the base field is the rational number field, for which the stronger question, and thus the extended abelian rank one Stark conjecture, have an affirmative answer