14 problems

The four-case integrality conjecture. In each of the following four cases, the assertions (a), (b), (c), and (d) are equivalent for every $xiin K1(R3):

Let $G$ be a compact $p$-adic Lie group, and let $\pi_\xi$ be an irreducible representation with representation dimension $d_\xi$. Consider the lower-bound property for the Hilbert…

Let $G$ be a compact nilpotent $p$-adic Lie group, let $\widehat G$ denote its unitary dual, and let $\pi_\xi$ be an irreducible representation of $G$ with representation dimension…

$\mathfrak{M}_H(G)$-conjecture. The Iwasawa module $X(E/F_\infty)$ belongs to the category $\mathfrak{M}_H(G)$. This is the non-commutative Iwasawa-theoretic expectation that arith…

Ardakov–Brown conjecture.

Characteristic-$p$ analytic cohomology conjecture. (a) If there is a sequence of subgroups $H_j$ of $H$ forming a neighborhood basis of the identity such that

Let $K$ be a $p$-adic field, let $K_\infty/K$ be an infinitely ramified $p$-adic Lie extension containing a cyclotomic extension, and let $d$ be the dimension of its Galois group…

Uniformity conjecture. Then $\Gamma$ must be uniform. The claim concerns the relationship between the algebraic generator invariant and the analytic dimension of a torsion-free pro…

Let ${\rm K}$ be a number field, and let $\Gamma$ be a uniform pro-$p$ group of dimension $d>2$, so that $\Gamma$ is infinite. A Galois extension ${\rm L}/{\rm K}$ is finitely and…

Let $K$ be a number field, let $M$ be a representation of $\operatorname{G}_K$ on a finite-dimensional vector space over $\mathbb{F}_p$, and let $\mathcal{L}$ be as in the precedin…

Noncommutative Iwasawa $\mathfrak{M}_{H}(G)$-conjecture. For totally real fields, $\mathcal{X}'$ should belong to $\mathfrak{M}_{H}(G)$.

Let $G$ be a ${\mathbb Q}_p$-analytic group over $L$, and let $\Pi$ be an absolutely irreducible admissible $L$-Banach space representation of $G$. Denote by $\Pi_{\rm an}$ the spa…

Let $G$ be a general compact $p$-adic Lie group and let $\mathbb{Z}_p[[G]]$ be its Iwasawa algebra. Define $SK_1(\mathbb{Z}_p[[G]])$ as the kernel of the reduced-norm or theta-type…

Let $p\neq 2$ be a prime, let $F$ be a totally real number field, and let $F^{\infty}/F$ be a totally real Galois extension of infinite degree containing the cyclotomic…