7 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…

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

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 $G^\Gamma$: