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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.

The goal of this paper is to show that the equivariant Tamagawa number conjecture implies the extended abelian Stark conjecture contained in [12] and [11]. In particular, this gives the first proof of the extended abelian Stark conjecture for the base field{\mathbb{Q}}, since the equivariant Tamagawa number conjecture away from 2 was proved in this context by Burns and Greither in [8] and Flach completed their results at 2 in [13] and [14].

Let ℓ be a rational prime. Previously, abelian ℓ-towers of multigraphs were introduced which are analogous to Z ℓ -extensions of number fields. It was shown that for a certain class of towers of bouquets, the growth of the ℓ-part of the number of spanning trees behaves in a predictable manner (analogous to a well-known theorem of Iwasawa for Z ℓ -extensions of number fields). In this paper, we give a generalization to a broader class of regular abelian ℓ-towers of bouquets than was originally considered. To carry this out, we observe that certain shifted Chebyshev polynomials are members of a continuously parametrized family of power series with coefficients in Z ℓ and then study the special value at s = 1 of the Artin-Ihara L-function ℓ-adically.

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