12 problems

Let $G$ be a compact $p$-adic analytic group. Write $\Omega_G$ for its completed group algebra over the coefficient ring used in the paper. Call $G$ dihedral-free if every orbital…

Let $G$ be a compact $p$-adic analytic group, let $H$ be a closed normal subgroup of $G$, and write $\Omega_{G,H}$ and $\Lambda_{G,H}$ for the corresponding Iwasawa algebras. For…

Let $G$ be a $p$-valuable group, let $K/\mathbb{Q}_p$ be a finite extension with ring of integers $\mathcal{O}$, and let $Z=Z(G)$ be the centre of $G$. For a prime ideal $P$ of…

Let $G$ be a $p$-valuable group, let $K/\mathbb{Q}_p$ be a finite extension with ring of integers $\mathcal{O}$, and let $P$ be a prime ideal of the Iwasawa algebra…

Finite-codimension conjecture. If $G$ is an almost simple uniform pro-$p$ group, then every non-zero prime ideal of $KG$ has finite codimension.

Ardakov–Brown conjecture.

Let $K$) be a finite extension of $\mathbb{Q}_p$ and let $\mathfrak{G}$ be a compact open subgroup of $G(R)$. Define the Iwasawa algebra … A primitive ideal of $K\mathfrak{G}$ is t…

The standardness conjecture. Let $G$ be a solvable, uniform pro-$p$ group, and let $P$ be a prime ideal in $\mathcal{O}G$. Then $P$ is virtually standard, and moreover if $p\in P$…

Standard-prime ideal conjecture. If $P$ is a prime ideal of $KG$, then $P$ is standard.

Let $L$ be either an arbitrary Galois extension $L^0$ of finite absolute ramification index over $\mathbb{Q}_p$ or the completion of such an extension, let $S=\mathcal{O}_L$, and l…

Fixed-point control conjecture. If $I$ is a $\Gamma$-invariant faithful prime ideal of $kG$, then $I$ is controlled by the subgroup of $\Gamma$-fixed points $G^\Gamma$:

Let $\Phi$ be a root system and let $G$ be a uniform pro-$p$ group of Chevalley type, with Iwasawa algebra $\Omega_G$ and Frobenius pair $(\Omega_G,\Omega_{G^p})$. The prime $p$ is…