12 problems

Brumer's conjecture. If $G$ is abelian, then

Coates–Sinnott conjecture. If $G$ is abelian, then

Let $L/K$ be an abelian extension of number fields with Galois group $G$. Let $S$ be a finite set of places of $K$ containing all archimedean places and all places ramified in…

Burns's conjecture. One has $\Theta(\phi)\in\mathbb{Z}[G]$, moreover $\Theta(\phi)\in\operatorname{Ann}_{\mathbb{Z}[G]}(Cl_S(K))$, and if $S'$ satisfies…

Let $F/K$ be an abelian CM extension of a totally real number field $K$, let ${\bf f}$ be the conductor of $F/K$, and let $\Theta_n({\bf b},{\bf f})$ be the higher Stickelberger el…

Let $K'$ be the relevant extension, let $H'$ be its Galois group, let $I(H')$ be the augmentation ideal, and let $r_{K'}$ be the rank of the $S(K')$-unit group. Let $theta_{H'}$ be…

Let $H/F$ be as in the Brumer–Stark setting, let $L/F$ be a finite abelian CM extension containing $H$, and write $\mathfrak{g}=\operatorname{Gal}(L/F)$ and…

Let $F$ be a totally real field and let $H/F$ be a finite abelian CM extension. Let $\theta_{S,T}(H/F,0)$ denote the appropriately smoothed Stickelberger element associated to…

With $K/F$, $G$, $V$, and $D'=\prod_{v\in V}I_{G_v,G}\subset\mathbb{Z}[G]$ as above, let $\Theta_{K,S,T}$ be the Stickelberger element. Order-of-vanishing conjecture. … This is the…

Finite-level Iwasawa Main Conjecture. The characteristic ideal of $mathcal A(mathcal F_d)$ is generated by the Stickelberger element:

Let $K/k$ be a finite Galois CM-extension of number fields with Galois group $G$. Let $S$ contain all infinite places and all finite places ramified in $K/k$, and let…

Let $k$ be totally real, let $K$ be totally real or a CM-field, let $K/k$ be an abelian extension with Galois group $G$, and let $S$ contain the infinite places and the places rami…