The non-abelian Brumer conjecture

Equivalent formulations 2

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

1. Non-abelian Brumer conjecture

   Let $K/k$ be a finite Galois CM-extension with Galois group $G$. Let $S$ contain all ramified finite places and all infinite places of $k$, let $\goth{A}_S$ be the module generated by the elements $\delta_T(0)$ for sets $T$ satisfying $Hyp(S,T)$, let $\mathcal I(G)$ be the relevant integrality module, and let $\mathcal H(G)$ be the associated denominator ideal. Non-abelian Brumer conjecture. One has

   $$\goth{A}_S\theta_S\subset\mathcal I(G),$$

   and, for every $x\in\mathcal H(G)$,

   $$x\cdot\goth{A}_S\theta_S\subset\operatorname{Ann}_{\mathbb ZG}(\mathrm{cl}_K).$$

   This conjecture generalizes Brumer's conjecture from abelian to non-abelian Galois groups. In the abelian case, the paper notes that it recovers Brumer's conjecture.

   source: Andreas Nickel, “Equivariant Iwasawa theory and non-abelian Stark-type conjectures”, arXiv:1109.5525 (2012).

2. Non-abelian Brumer conjecture

   Let $L/K$ be a Galois CM-extension with Galois group $G$, and let $S$ be a finite set of places of $K$ containing all archimedean places and all places ramified in $L/K$. Let $\mathcal H(G)$, $\mathfrak A_S$, and $\theta_S$ be the auxiliary ideal, integrality module, and Stickelberger element defined in the paper. Non-abelian Brumer conjecture. For every $x\in\mathcal H(G)$,

   $$x\cdot\mathfrak A_S\theta_S\subseteq\mathrm{Ann}_{\zeta(\mathbb Z[G])}(\mathrm{cl}_L).$$

   This generalizes Brumer's conjecture from abelian to arbitrary Galois extensions. It is known in various special cases and follows from stronger refined conjectures in settings treated in the literature, but remains open in general.

   source: Andreas Nickel, “Conjectures of Brumer, Gross and Stark”, arXiv:1707.04432 (2017).