The non-abelian Brumer–Stark 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-Stark 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 $\omega_K=\operatorname{nr}(|\mu_K|)$, and let $\mathcal H(G)$ be the associated denominator ideal. For $\alpha\in K^{\times}$, let $S_\alpha$ be the finite places whose primes divide $N_{K/k}(\alpha)$; call $\alpha$ an anti-unit when $\alpha^{1+j}=1$. Non-abelian Brumer-Stark conjecture. For every $x\in\mathcal H(G)$ and every fractional ideal $\mathfrak a$ of $K$, there is an anti-unit $\alpha=\alpha(x,\mathfrak a,S)\in K^{\times}$ such that

   $$\mathfrak a^{x\cdot\omega_K\cdot\theta_S}=(\alpha),$$

   and, for every finite set $T$ such that $Hyp(S\cup S_\alpha,T)$ holds, there is an $\alpha_T\in E_{S_\alpha}^T$ satisfying

   $$\alpha^{z\cdot\delta_T(0)}=\alpha_T^{z\cdot\omega_K}$$

   for every $z\in\mathcal H(G)$. This is the non-abelian refinement of the Brumer-Stark conjecture, strengthening ideal annihilation to the principality of explicitly acted-on ideals and compatibility with ray-unit data.

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

2. Non-abelian Brumer–Stark 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 $\mathcal H(G)$, $\omega_L$, $\theta_S$, $E_{L,S_\epsilon}^T$, and $\mathsf{Hyp}(S\cup S_\epsilon,T)$ have the meanings defined in the paper; an anti-unit satisfies $\epsilon^{1+j}=1$. Non-abelian Brumer–Stark conjecture. For every $x\in\mathcal H(G)$,

   $$x\omega_L\theta_S\in\zeta(\mathbb Z[G]).$$

   Moreover, for every non-zero fractional ideal $\mathfrak a$ of $L$, there is an anti-unit $\epsilon=\epsilon(x,\mathfrak a,S)\in L^\times$ with

   $$\mathfrak a^{x\omega_L\theta_S}=(\epsilon),$$

   and, for every finite set $T$ satisfying $\mathsf{Hyp}(S\cup S_\epsilon,T)$, an $\epsilon_T\in E_{L,S_\epsilon}^T$ such that

   $$\epsilon^{z\delta_T(0)}=\epsilon_T^{z\omega_L}$$

   for each $z\in\mathcal H(G)$. This is a refined non-abelian form of Brumer–Stark, imposing both principality and explicit unit relations. It is open in general.

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