9 problems

Let $p$ be a prime, and let $n,k\geq 1$. Consider the general linear groups over the truncated polynomial ring and the residue ring … For a finite group $G$, define its representat…

Let $G$ be a FAb compact $p$-adic analytic group. Its representation zeta function $zeta_G(s)$ is defined by … where $r_n(G)$ is the number of isomorphism classes of continuous com…

Let $\mathbb{Q}[\varepsilon_n]=\mathbb{Q}[x]/(x^n)$, and let $\mathbf{H}[\varepsilon_n]$ denote the group scheme obtained from the Heisenberg Lie algebra tensored with…

Let $\mathcal{O}$ be the ring of integers of a number field $K$, and let $\mathbf{G}$ be a unipotent group scheme arising from a finitely generated free torsion-free $\mathcal{O}$-…

Let $R$ be a ring which is torsion-free and finitely generated over $\mathbb{Z}$, and let $\mathbf{H}(R)$ be the corresponding Heisenberg group. Write $\alpha(\mathbf{H})$ for the…

Let $K$ be the number field with ring of integers $O$, and let $H$ be the Heisenberg group scheme. For $ninmathbb{N}$, write $zeta{mathbf{H}(mathcal{O}[x]/(x^n))}(s)$ for its rep…

Let $n\in\mathbb{N}_{\geq 2}$. For a divisor $\iota(v)=\gcd(q_v-\varepsilon(v),n)$ of $n$, where $\varepsilon(v)\in\{1,-1\}$ records whether the relevant quadratic extension splits…

Rational representation zeta function conjecture. Every FAb compact $2$-adic Lie group has $\mathrm{r.r.z.f.}_{(2)}$.

The representation zeta function conjecture. The zeta function is