Theoretical evidence in support of the extended abelian rank one Stark conjecture for the base field Q

Abstract

The main goal of this paper is to study two infinite families of abelian extensions of Q, one with Galois groups of the type Z/2Z × Z/2Z × Z/mZ, where m is an odd integer, and one with Galois groups of the type Z/2Z × Z/ Z × Z/ Z, where is an odd prime. We show partial results towards the extended abelian rank one Stark conjecture for the first family and we show that the extended abelian rank one Stark conjecture is true for the second one. Contents 1. Introduction 1 Acknowledgement 2 1.1. Notation 2 1.2. Background 2 2. A generalization of the Brumer-Stark conjecture 3 2.1. Local version of the extension of the Brumer-Stark conjecture 5 2.2. Further remarks 8 2.3. Non semi-simplicity 9 3. Two infinite families of abelian extensions of Q 3.1. The first family of abelian extensions of Q to be studied 10 The ramified prime p. The unramified prime. 11 3.2. The second family of abelian extensions of Q to be studied The ramified primes p 1 and p 2. 13 The unramified primes p 3 ,. .. , p +1. 13 4. Conclusion References