On a generalization of the rank one Rubin–Stark conjecture

Abstract

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.

Numerical evidence for higher order Stark-type conjectures

“…That is, if K/k is a finite abelian extension of number fields such that Hypothesis 3.15 is satisfied for r = 1, then there exists an S K -unit ε 0 ∈ E S (K) satisfying (1) e ′ 1,S · ε 0 = ε 0 in QE S (K), Such an S K -unit is called a Stark unit and is unique up to a root of unity. If necessary, see §3.8 of [18] for a comparison between the various slightly different formulations of Stark's abelian rank one conjecture that one can find in the literature. Remark 3.24.…”

“…, t. In order to do so, we use the following well-known lemma: Proof. See Lemma 4.33 of [18] for details if needed.…”

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

“…In this paper, we will be concerned almost exclusively with the case where v ∈ S min is a finite prime. In this case, as explained in [15], it is possible to formulate an extension of the classical Brumer-Stark conjecture which, if true, implies that St(K/k, S, v) has an affirmative answer. This extension of the Date: July 22, 2012.…”

“…The extended abelian rank one Stark conjecture was stated for the first time in [3]. Our approach is based on [15] and we refer to this latter paper for the statement of the conjecture (Conjecture 3.6). See also [2] where arbitrary orders of vanishing are treated.…”

“…Here, we work exclusively with the S-version of the conjecture. For a given 1-cover S (see §1.2 below for the definitions of a 1-cover and S min ), rather than studying the extended abelian rank one Stark conjecture, we study Question 4.2 of [15] (denoted by St(K/k, S, v) for a given 1-cover S and a place v ∈ S min ). It was shown that an affirmative answer to St(K/k, S, v) for all v ∈ S min implies the extended abelian rank one Stark conjecture (see Proposition 4.3 of [15]).…”