6 problems

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 $n\in\mathbb{N}$. For $\varepsilon\in\{-1,1\}$, let $\mathfrak{o}$ be a compact discrete valuation ring with residue cardinality $q$, let $\mathsf{G}$ be one of the…

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…

The representation zeta function conjecture. The zeta function is